Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 93.1%

Time: 9.4s

Alternatives: 22

Speedup: 9.6×

• 39.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.9% 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: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;l\_m \leq 2 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{t} \cdot \left(\frac{l\_m}{k} \cdot \frac{\frac{l\_m}{t\_1}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot t} \cdot \left(\frac{\cos k \cdot l\_m}{k} \cdot \frac{l\_m}{k}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= l_m 2e-162)
     (* (/ 2.0 t) (* (/ l_m k) (/ (/ l_m t_1) k)))
     (* (/ 2.0 (* t_1 t)) (* (/ (* (cos k) l_m) k) (/ l_m k))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (l_m <= 2e-162) {
		tmp = (2.0 / t) * ((l_m / k) * ((l_m / t_1) / k));
	} else {
		tmp = (2.0 / (t_1 * t)) * (((cos(k) * l_m) / k) * (l_m / k));
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (l_m <= 2d-162) then
        tmp = (2.0d0 / t) * ((l_m / k) * ((l_m / t_1) / k))
    else
        tmp = (2.0d0 / (t_1 * t)) * (((cos(k) * l_m) / k) * (l_m / k))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (l_m <= 2e-162) {
		tmp = (2.0 / t) * ((l_m / k) * ((l_m / t_1) / k));
	} else {
		tmp = (2.0 / (t_1 * t)) * (((Math.cos(k) * l_m) / k) * (l_m / k));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if l_m <= 2e-162:
		tmp = (2.0 / t) * ((l_m / k) * ((l_m / t_1) / k))
	else:
		tmp = (2.0 / (t_1 * t)) * (((math.cos(k) * l_m) / k) * (l_m / k))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (l_m <= 2e-162)
		tmp = Float64(Float64(2.0 / t) * Float64(Float64(l_m / k) * Float64(Float64(l_m / t_1) / k)));
	else
		tmp = Float64(Float64(2.0 / Float64(t_1 * t)) * Float64(Float64(Float64(cos(k) * l_m) / k) * Float64(l_m / k)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (l_m <= 2e-162)
		tmp = (2.0 / t) * ((l_m / k) * ((l_m / t_1) / k));
	else
		tmp = (2.0 / (t_1 * t)) * (((cos(k) * l_m) / k) * (l_m / k));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l$95$m, 2e-162], N[(N[(2.0 / t), $MachinePrecision] * N[(N[(l$95$m / k), $MachinePrecision] * N[(N[(l$95$m / t$95$1), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.99999999999999991e-162

   1. Initial program 29.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 67.8

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

   5. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 59.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. associate-/l* N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-pow.f64 79.1

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

   12. Applied rewrites 79.1%

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

   if 1.99999999999999991e-162 < l

   1. Initial program 41.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 98.4%

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

Alternative 2: 92.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sin k \cdot k\\ \frac{2}{t} \cdot \left(\frac{\cos k \cdot l\_m}{t\_1} \cdot \frac{l\_m}{t\_1}\right) \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (sin k) k)))
   (* (/ 2.0 t) (* (/ (* (cos k) l_m) t_1) (/ l_m t_1)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = sin(k) * k;
	return (2.0 / t) * (((cos(k) * l_m) / t_1) * (l_m / t_1));
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    t_1 = sin(k) * k
    code = (2.0d0 / t) * (((cos(k) * l_m) / t_1) * (l_m / t_1))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.sin(k) * k;
	return (2.0 / t) * (((Math.cos(k) * l_m) / t_1) * (l_m / t_1));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.sin(k) * k
	return (2.0 / t) * (((math.cos(k) * l_m) / t_1) * (l_m / t_1))

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(sin(k) * k)
	return Float64(Float64(2.0 / t) * Float64(Float64(Float64(cos(k) * l_m) / t_1) * Float64(l_m / t_1)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	t_1 = sin(k) * k;
	tmp = (2.0 / t) * (((cos(k) * l_m) / t_1) * (l_m / t_1));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]}, N[(N[(2.0 / t), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l$95$m / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 75.6

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

5. Applied rewrites 75.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. associate-*l/ N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. times-frac N/A

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

   15. metadata-eval N/A

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

   16. associate-*r/ N/A

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

7. Applied rewrites 74.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. pow2 N/A

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

   10. associate-/l/ N/A

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

   11. associate-*l* N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

9. Applied rewrites 94.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-cos.f64 N/A

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

    7. frac-times N/A

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

    8. *-commutative N/A

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

    9. associate-*l* N/A

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

    10. pow2 N/A

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

    11. *-commutative N/A

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

    12. pow2 N/A

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

    13. lift-pow.f64 N/A

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

    14. lift-sin.f64 N/A

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

    15. associate-/r* N/A

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

    16. *-commutative N/A

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

    17. unpow-prod-down N/A

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

    18. unpow2 N/A

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

11. Applied rewrites 94.5%

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

Alternative 3: 89.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \cos k \cdot l\_m\\ \mathbf{if}\;k \leq 1.75 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{t\_1}{\left(k \cdot k\right) \cdot t} \cdot \frac{l\_m}{{\sin k}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{\frac{l\_m}{k} \cdot \frac{t\_1}{k}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (cos k) l_m)))
   (if (<= k 1.75e+39)
     (* (* (/ t_1 (* (* k k) t)) (/ l_m (pow (sin k) 2.0))) 2.0)
     (*
      (/ 2.0 t)
      (/ (* (/ l_m k) (/ t_1 k)) (- 0.5 (* 0.5 (cos (* 2.0 k)))))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = cos(k) * l_m;
	double tmp;
	if (k <= 1.75e+39) {
		tmp = ((t_1 / ((k * k) * t)) * (l_m / pow(sin(k), 2.0))) * 2.0;
	} else {
		tmp = (2.0 / t) * (((l_m / k) * (t_1 / k)) / (0.5 - (0.5 * cos((2.0 * k)))));
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l_m
    if (k <= 1.75d+39) then
        tmp = ((t_1 / ((k * k) * t)) * (l_m / (sin(k) ** 2.0d0))) * 2.0d0
    else
        tmp = (2.0d0 / t) * (((l_m / k) * (t_1 / k)) / (0.5d0 - (0.5d0 * cos((2.0d0 * k)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.cos(k) * l_m;
	double tmp;
	if (k <= 1.75e+39) {
		tmp = ((t_1 / ((k * k) * t)) * (l_m / Math.pow(Math.sin(k), 2.0))) * 2.0;
	} else {
		tmp = (2.0 / t) * (((l_m / k) * (t_1 / k)) / (0.5 - (0.5 * Math.cos((2.0 * k)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.cos(k) * l_m
	tmp = 0
	if k <= 1.75e+39:
		tmp = ((t_1 / ((k * k) * t)) * (l_m / math.pow(math.sin(k), 2.0))) * 2.0
	else:
		tmp = (2.0 / t) * (((l_m / k) * (t_1 / k)) / (0.5 - (0.5 * math.cos((2.0 * k)))))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (k <= 1.75e+39)
		tmp = Float64(Float64(Float64(t_1 / Float64(Float64(k * k) * t)) * Float64(l_m / (sin(k) ^ 2.0))) * 2.0);
	else
		tmp = Float64(Float64(2.0 / t) * Float64(Float64(Float64(l_m / k) * Float64(t_1 / k)) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = cos(k) * l_m;
	tmp = 0.0;
	if (k <= 1.75e+39)
		tmp = ((t_1 / ((k * k) * t)) * (l_m / (sin(k) ^ 2.0))) * 2.0;
	else
		tmp = (2.0 / t) * (((l_m / k) * (t_1 / k)) / (0.5 - (0.5 * cos((2.0 * k)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[k, 1.75e+39], N[(N[(N[(t$95$1 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7500000000000001e39

   1. Initial program 34.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.7%

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

   if 1.7500000000000001e39 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l/ N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

   9. Applied rewrites 95.4%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 95.1

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

   11. Applied rewrites 95.1%

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

Alternative 4: 82.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-5) then
        tmp = ((2.0d0 / t) / k) * ((l_m * (l_m / (sin(k) ** 2.0d0))) / k)
    else
        tmp = (2.0d0 / t) * (((l_m / k) * ((cos(k) * l_m) / k)) / (0.5d0 - (0.5d0 * cos((2.0d0 * k)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 7e-5) {
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / Math.pow(Math.sin(k), 2.0))) / k);
	} else {
		tmp = (2.0 / t) * (((l_m / k) * ((Math.cos(k) * l_m) / k)) / (0.5 - (0.5 * Math.cos((2.0 * k)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 7e-5:
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / math.pow(math.sin(k), 2.0))) / k)
	else:
		tmp = (2.0 / t) * (((l_m / k) * ((math.cos(k) * l_m) / k)) / (0.5 - (0.5 * math.cos((2.0 * k)))))
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 7e-5)
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / (sin(k) ^ 2.0))) / k);
	else
		tmp = (2.0 / t) * (((l_m / k) * ((cos(k) * l_m) / k)) / (0.5 - (0.5 * cos((2.0 * k)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 72.9

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

   5. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 65.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r/ N/A

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

       6. pow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 67.3

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

   12. Applied rewrites 82.3%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 31.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 85.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l/ N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

   9. Applied rewrites 96.2%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 95.9

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

   11. Applied rewrites 95.9%

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

Alternative 5: 82.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{2}{t}}{k} \cdot \frac{l\_m \cdot \frac{l\_m}{{\sin k}^{2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\cos k \cdot l\_m}{k} \cdot \frac{l\_m}{k}\right) \cdot 4}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 7e-5)
   (* (/ (/ 2.0 t) k) (/ (* l_m (/ l_m (pow (sin k) 2.0))) k))
   (/
    (* (* (/ (* (cos k) l_m) k) (/ l_m k)) 4.0)
    (* (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t) 2.0))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 7e-5) {
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / pow(sin(k), 2.0))) / k);
	} else {
		tmp = ((((cos(k) * l_m) / k) * (l_m / k)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k)))) * t) * 2.0);
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-5) then
        tmp = ((2.0d0 / t) / k) * ((l_m * (l_m / (sin(k) ** 2.0d0))) / k)
    else
        tmp = ((((cos(k) * l_m) / k) * (l_m / k)) * 4.0d0) / (((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * t) * 2.0d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 7e-5) {
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / Math.pow(Math.sin(k), 2.0))) / k);
	} else {
		tmp = ((((Math.cos(k) * l_m) / k) * (l_m / k)) * 4.0) / (((0.5 - (0.5 * Math.cos((2.0 * k)))) * t) * 2.0);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 7e-5:
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / math.pow(math.sin(k), 2.0))) / k)
	else:
		tmp = ((((math.cos(k) * l_m) / k) * (l_m / k)) * 4.0) / (((0.5 - (0.5 * math.cos((2.0 * k)))) * t) * 2.0)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 7e-5)
		tmp = Float64(Float64(Float64(2.0 / t) / k) * Float64(Float64(l_m * Float64(l_m / (sin(k) ^ 2.0))) / k));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k) * l_m) / k) * Float64(l_m / k)) * 4.0) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t) * 2.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 7e-5)
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / (sin(k) ^ 2.0))) / k);
	else
		tmp = ((((cos(k) * l_m) / k) * (l_m / k)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k)))) * t) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 7e-5], N[(N[(N[(2.0 / t), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 72.9

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

   5. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 65.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r/ N/A

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

       6. pow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 67.3

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

   12. Applied rewrites 82.3%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 31.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 96.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

         \[\leadsto \frac{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot 4}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot 2} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot 4}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot 4}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot 2} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot 4}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot 2} \]

      8. lower-*.f64 95.9

         \[\leadsto \frac{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot 4}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot 2} \]

   9. Applied rewrites 95.9%

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

Alternative 6: 77.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-5) then
        tmp = ((2.0d0 / t) / k) * ((l_m * (l_m / (sin(k) ** 2.0d0))) / k)
    else
        tmp = (2.0d0 / t) * ((((cos(k) * l_m) * l_m) / (0.5d0 - (0.5d0 * cos((2.0d0 * k))))) / (k * k))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 7e-5) {
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / Math.pow(Math.sin(k), 2.0))) / k);
	} else {
		tmp = (2.0 / t) * ((((Math.cos(k) * l_m) * l_m) / (0.5 - (0.5 * Math.cos((2.0 * k))))) / (k * k));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 7e-5:
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / math.pow(math.sin(k), 2.0))) / k)
	else:
		tmp = (2.0 / t) * ((((math.cos(k) * l_m) * l_m) / (0.5 - (0.5 * math.cos((2.0 * k))))) / (k * k))
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 7e-5)
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / (sin(k) ^ 2.0))) / k);
	else
		tmp = (2.0 / t) * ((((cos(k) * l_m) * l_m) / (0.5 - (0.5 * cos((2.0 * k))))) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 72.9

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

   5. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 65.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r/ N/A

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

       6. pow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 67.3

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

   12. Applied rewrites 82.3%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 31.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 85.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 85.1

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

   9. Applied rewrites 85.1%

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

Alternative 7: 77.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-5) then
        tmp = ((2.0d0 / t) / k) * ((l_m * (l_m / (sin(k) ** 2.0d0))) / k)
    else
        tmp = (2.0d0 / ((k * k) * t)) * ((cos(k) * (l_m * l_m)) / (0.5d0 - (0.5d0 * cos((2.0d0 * k)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 7e-5) {
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / Math.pow(Math.sin(k), 2.0))) / k);
	} else {
		tmp = (2.0 / ((k * k) * t)) * ((Math.cos(k) * (l_m * l_m)) / (0.5 - (0.5 * Math.cos((2.0 * k)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 7e-5:
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / math.pow(math.sin(k), 2.0))) / k)
	else:
		tmp = (2.0 / ((k * k) * t)) * ((math.cos(k) * (l_m * l_m)) / (0.5 - (0.5 * math.cos((2.0 * k)))))
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 7e-5)
		tmp = ((2.0 / t) / k) * ((l_m * (l_m / (sin(k) ^ 2.0))) / k);
	else
		tmp = (2.0 / ((k * k) * t)) * ((cos(k) * (l_m * l_m)) / (0.5 - (0.5 * cos((2.0 * k)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 72.9

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

   5. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 65.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r/ N/A

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

       6. pow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 67.3

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

   12. Applied rewrites 82.3%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 31.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 83.7

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

   7. Applied rewrites 83.7%

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

Alternative 8: 74.9% accurate, 1.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ 2.0 t) (* (/ l_m k) (/ (/ l_m (pow (sin k) 2.0)) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 / t) * ((l_m / k) * ((l_m / pow(sin(k), 2.0)) / k));
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 / t) * ((l_m / k) * ((l_m / (sin(k) ** 2.0d0)) / k))
end function

Java

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

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 / t) * ((l_m / k) * ((l_m / math.pow(math.sin(k), 2.0)) / k))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 / t) * Float64(Float64(l_m / k) * Float64(Float64(l_m / (sin(k) ^ 2.0)) / k)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 / t) * ((l_m / k) * ((l_m / (sin(k) ^ 2.0)) / k));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 / t), $MachinePrecision] * N[(N[(l$95$m / k), $MachinePrecision] * N[(N[(l$95$m / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 75.6

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

5. Applied rewrites 75.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. associate-*l/ N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. times-frac N/A

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

   15. metadata-eval N/A

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

   16. associate-*r/ N/A

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

7. Applied rewrites 74.9%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 65.5%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. lift-sin.f64 N/A

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

    7. associate-/l* N/A

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

    8. times-frac N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 N/A

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

    11. lower-/.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. lift-sin.f64 N/A

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

    14. lift-pow.f64 78.4

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

12. Applied rewrites 78.4%

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

Alternative 9: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{\sin k \cdot k}\\ \frac{2}{t} \cdot \left(t\_1 \cdot t\_1\right) \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (/ l_m (* (sin k) k)))) (* (/ 2.0 t) (* t_1 t_1))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = l_m / (sin(k) * k);
	return (2.0 / t) * (t_1 * t_1);
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    t_1 = l_m / (sin(k) * k)
    code = (2.0d0 / t) * (t_1 * t_1)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = l_m / (Math.sin(k) * k);
	return (2.0 / t) * (t_1 * t_1);
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = l_m / (math.sin(k) * k)
	return (2.0 / t) * (t_1 * t_1)

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(l_m / Float64(sin(k) * k))
	return Float64(Float64(2.0 / t) * Float64(t_1 * t_1))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	t_1 = l_m / (sin(k) * k);
	tmp = (2.0 / t) * (t_1 * t_1);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(l$95$m / N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]}, N[(N[(2.0 / t), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 75.6

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

5. Applied rewrites 75.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. associate-*l/ N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. times-frac N/A

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

   15. metadata-eval N/A

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

   16. associate-*r/ N/A

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

7. Applied rewrites 74.9%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 65.5%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. pow2 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. lift-sin.f64 N/A

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

    7. associate-/l/ N/A

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

    8. lift-*.f64 N/A

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

    9. unpow-prod-down N/A

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

    10. unpow2 N/A

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

    11. times-frac N/A

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

    12. lower-*.f64 N/A

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

    13. lower-/.f64 N/A

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

    14. lift-sin.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. lower-/.f64 N/A

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

12. Applied rewrites 78.4%

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

Alternative 10: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5 \cdot 10^{-153}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{\frac{l\_m}{k} \cdot \frac{l\_m}{k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{l\_m \cdot l\_m}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 5e-153)
   (* (/ 2.0 t) (/ (* (/ l_m k) (/ l_m k)) (* k k)))
   (* (/ 2.0 (* (* k k) t)) (/ (* l_m l_m) (pow (sin k) 2.0)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 5e-153) {
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	} else {
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l_m <= 5d-153) then
        tmp = (2.0d0 / t) * (((l_m / k) * (l_m / k)) / (k * k))
    else
        tmp = (2.0d0 / ((k * k) * t)) * ((l_m * l_m) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 5e-153) {
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	} else {
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if l_m <= 5e-153:
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k))
	else:
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 5e-153)
		tmp = Float64(Float64(2.0 / t) * Float64(Float64(Float64(l_m / k) * Float64(l_m / k)) / Float64(k * k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l_m * l_m) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (l_m <= 5e-153)
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	else
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 5e-153], N[(N[(2.0 / t), $MachinePrecision] * N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000033e-153

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.0

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

   5. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 59.6%

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

   11. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-/.f64 76.8

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

   13. Applied rewrites 76.8%

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

   if 5.00000000000000033e-153 < l

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 88.9

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}} \]

      2. lift-*.f64 77.6

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}} \]

   8. Applied rewrites 77.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\color{blue}{\sin k}}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{\frac{l\_m \cdot \frac{l\_m}{{\sin k}^{2}}}{k} \cdot 2}{k \cdot t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ (* (/ (* l_m (/ l_m (pow (sin k) 2.0))) k) 2.0) (* k t)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (((l_m * (l_m / pow(sin(k), 2.0))) / k) * 2.0) / (k * t);
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (((l_m * (l_m / (sin(k) ** 2.0d0))) / k) * 2.0d0) / (k * t)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (((l_m * (l_m / Math.pow(Math.sin(k), 2.0))) / k) * 2.0) / (k * t);
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (((l_m * (l_m / math.pow(math.sin(k), 2.0))) / k) * 2.0) / (k * t)

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(Float64(Float64(l_m * Float64(l_m / (sin(k) ^ 2.0))) / k) * 2.0) / Float64(k * t))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (((l_m * (l_m / (sin(k) ^ 2.0))) / k) * 2.0) / (k * t);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(N[(N[(l$95$m * N[(l$95$m / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 75.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. associate-*l/ N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

   14. times-frac N/A

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

   16. associate-*r/ N/A

       \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

7. Applied rewrites 74.9%

   \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
9. Step-by-step derivation

10. Applied rewrites 65.5%

    \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \color{blue}{\frac{2}{t}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \frac{2}{t} \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \frac{\color{blue}{2}}{t} \]

    5. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k}}{k} \cdot \frac{\color{blue}{2}}{t} \]

    6. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k}}{k} \cdot \frac{2}{\color{blue}{t}} \]

    7. frac-times N/A

       \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k} \cdot 2}{\color{blue}{k \cdot t}} \]

    8. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k} \cdot 2}{\color{blue}{k \cdot t}} \]

12. Applied rewrites 78.2%

    \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{\sin k}^{2}}}{k} \cdot 2}{\color{blue}{k \cdot t}} \]
13. Add Preprocessing

Alternative 12: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{t} \cdot \left(\frac{l\_m}{{\sin k}^{2}} \cdot \frac{l\_m}{k \cdot k}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ 2.0 t) (* (/ l_m (pow (sin k) 2.0)) (/ l_m (* k k)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 / t) * ((l_m / pow(sin(k), 2.0)) * (l_m / (k * k)));
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 / t) * ((l_m / (sin(k) ** 2.0d0)) * (l_m / (k * k)))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (2.0 / t) * ((l_m / Math.pow(Math.sin(k), 2.0)) * (l_m / (k * k)));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 / t) * ((l_m / math.pow(math.sin(k), 2.0)) * (l_m / (k * k)))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 / t) * Float64(Float64(l_m / (sin(k) ^ 2.0)) * Float64(l_m / Float64(k * k))))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 / t) * ((l_m / (sin(k) ^ 2.0)) * (l_m / (k * k)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 / t), $MachinePrecision] * N[(N[(l$95$m / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 75.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. associate-*l/ N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

   14. times-frac N/A

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

   16. associate-*r/ N/A

       \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

7. Applied rewrites 74.9%

   \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
9. Step-by-step derivation

10. Applied rewrites 65.5%

    \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot \color{blue}{k}} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{\color{blue}{k \cdot k}} \]

    3. pow2 N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{\color{blue}{2}}} \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{\color{blue}{k}}^{2}} \]

    5. lift-pow.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \]

    6. lift-sin.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \]

    7. associate-/l/ N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

    8. lift-*.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2}} \cdot {k}^{2}} \]

    9. times-frac N/A

       \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right) \]

    11. lower-/.f64 N/A

        \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\color{blue}{\ell}}{{k}^{2}}\right) \]

    12. lift-sin.f64 N/A

        \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{{k}^{2}}\right) \]

    13. lift-pow.f64 N/A

        \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{{k}^{2}}\right) \]

    14. lower-/.f64 N/A

        \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{\color{blue}{{k}^{2}}}\right) \]

    15. pow2 N/A

        \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{k \cdot \color{blue}{k}}\right) \]

    16. lift-*.f64 77.8

        \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell}{k \cdot \color{blue}{k}}\right) \]

12. Applied rewrites 77.8%

    \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]
13. Add Preprocessing

Alternative 13: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{\frac{l\_m}{k} \cdot \frac{l\_m}{k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(l\_m \cdot l\_m\right) \cdot 2}{{\left(\sin k \cdot k\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= (* l_m l_m) 2e-21)
   (* (/ 2.0 t) (/ (* (/ l_m k) (/ l_m k)) (* k k)))
   (/ (* (* l_m l_m) 2.0) (* (pow (* (sin k) k) 2.0) t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if ((l_m * l_m) <= 2e-21) {
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	} else {
		tmp = ((l_m * l_m) * 2.0) / (pow((sin(k) * k), 2.0) * t);
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l_m * l_m) <= 2d-21) then
        tmp = (2.0d0 / t) * (((l_m / k) * (l_m / k)) / (k * k))
    else
        tmp = ((l_m * l_m) * 2.0d0) / (((sin(k) * k) ** 2.0d0) * t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if ((l_m * l_m) <= 2e-21) {
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	} else {
		tmp = ((l_m * l_m) * 2.0) / (Math.pow((Math.sin(k) * k), 2.0) * t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if (l_m * l_m) <= 2e-21:
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k))
	else:
		tmp = ((l_m * l_m) * 2.0) / (math.pow((math.sin(k) * k), 2.0) * t)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 2e-21)
		tmp = Float64(Float64(2.0 / t) * Float64(Float64(Float64(l_m / k) * Float64(l_m / k)) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l_m * l_m) * 2.0) / Float64((Float64(sin(k) * k) ^ 2.0) * t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if ((l_m * l_m) <= 2e-21)
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	else
		tmp = ((l_m * l_m) * 2.0) / (((sin(k) * k) ^ 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e-21], N[(N[(2.0 / t), $MachinePrecision] * N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999982e-21

   1. Initial program 28.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 70.0

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

      16. associate-*r/ N/A

          \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

   7. Applied rewrites 67.8%

      \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.2%

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]

   11. Taylor expanded in k around 0

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{k} \cdot k} \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{k \cdot k} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{k \cdot k} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

       6. lift-/.f64 90.2

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

   13. Applied rewrites 90.2%

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{k} \cdot k} \]

   if 1.99999999999999982e-21 < (*.f64 l l)

   1. Initial program 38.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 80.8

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

      16. associate-*r/ N/A

          \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

   7. Applied rewrites 81.6%

      \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.8%

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \color{blue}{\frac{2}{t}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \frac{2}{t} \]

       4. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \frac{\color{blue}{2}}{t} \]

       5. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \cdot \frac{2}{t} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \cdot \frac{2}{t} \]

       7. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \cdot \frac{2}{t} \]

       8. lift-sin.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \cdot \frac{2}{t} \]

       9. associate-/l/ N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot {k}^{2}} \cdot \frac{\color{blue}{2}}{t} \]

       10. unpow-prod-down N/A

           \[\leadsto \frac{\ell \cdot \ell}{{\left(\sin k \cdot k\right)}^{2}} \cdot \frac{2}{t} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{\ell \cdot \ell}{{\left(\sin k \cdot k\right)}^{2}} \cdot \frac{2}{\color{blue}{t}} \]

       12. frac-times N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \]

   12. Applied rewrites 66.1%

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{{\left(\sin k \cdot k\right)}^{2} \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 72.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{\left(l\_m \cdot \frac{l\_m}{{\sin k}^{2}}\right) \cdot 2}{\left(k \cdot k\right) \cdot t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ (* (* l_m (/ l_m (pow (sin k) 2.0))) 2.0) (* (* k k) t)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return ((l_m * (l_m / pow(sin(k), 2.0))) * 2.0) / ((k * k) * t);
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = ((l_m * (l_m / (sin(k) ** 2.0d0))) * 2.0d0) / ((k * k) * t)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return ((l_m * (l_m / Math.pow(Math.sin(k), 2.0))) * 2.0) / ((k * k) * t);
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return ((l_m * (l_m / math.pow(math.sin(k), 2.0))) * 2.0) / ((k * k) * t)

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(Float64(l_m * Float64(l_m / (sin(k) ^ 2.0))) * 2.0) / Float64(Float64(k * k) * t))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = ((l_m * (l_m / (sin(k) ^ 2.0))) * 2.0) / ((k * k) * t);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(N[(l$95$m * N[(l$95$m / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 75.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. associate-*l/ N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

   14. times-frac N/A

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

   16. associate-*r/ N/A

       \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

7. Applied rewrites 74.9%

   \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
9. Step-by-step derivation

10. Applied rewrites 65.5%

    \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \color{blue}{\frac{2}{t}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \frac{2}{t} \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \cdot \frac{\color{blue}{2}}{t} \]

    5. pow2 N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \cdot \frac{2}{t} \]

    6. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{{k}^{2}} \cdot \frac{2}{\color{blue}{t}} \]

    7. frac-times N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}} \cdot 2}{\color{blue}{{k}^{2} \cdot t}} \]

    8. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}} \cdot 2}{\color{blue}{{k}^{2} \cdot t}} \]

12. Applied rewrites 77.1%

    \[\leadsto \frac{\left(\ell \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot 2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
13. Add Preprocessing

Alternative 15: 72.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 2200:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{l\_m}{k} \cdot \frac{l\_m}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{\frac{\left(\cos k \cdot l\_m\right) \cdot l\_m}{k \cdot k}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 2200.0)
   (* (/ 2.0 (* (* k k) t)) (* (/ l_m k) (/ l_m k)))
   (* (/ 2.0 t) (/ (/ (* (* (cos k) l_m) l_m) (* k k)) (* k k)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 2200.0) {
		tmp = (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k));
	} else {
		tmp = (2.0 / t) * ((((cos(k) * l_m) * l_m) / (k * k)) / (k * k));
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2200.0d0) then
        tmp = (2.0d0 / ((k * k) * t)) * ((l_m / k) * (l_m / k))
    else
        tmp = (2.0d0 / t) * ((((cos(k) * l_m) * l_m) / (k * k)) / (k * k))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 2200.0) {
		tmp = (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k));
	} else {
		tmp = (2.0 / t) * ((((Math.cos(k) * l_m) * l_m) / (k * k)) / (k * k));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 2200.0:
		tmp = (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k))
	else:
		tmp = (2.0 / t) * ((((math.cos(k) * l_m) * l_m) / (k * k)) / (k * k))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 2200.0)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l_m / k) * Float64(l_m / k)));
	else
		tmp = Float64(Float64(2.0 / t) * Float64(Float64(Float64(Float64(cos(k) * l_m) * l_m) / Float64(k * k)) / Float64(k * k)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 2200.0)
		tmp = (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k));
	else
		tmp = (2.0 / t) * ((((cos(k) * l_m) * l_m) / (k * k)) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 2200.0], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2200

   1. Initial program 34.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 73.1

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lower-/.f64 79.6

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied rewrites 79.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

   if 2200 < k

   1. Initial program 31.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 83.6

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

      16. associate-*r/ N/A

          \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{k}^{2}}}{k \cdot k} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k}}{k \cdot k} \]

      2. lift-*.f64 63.7

         \[\leadsto \frac{2}{t} \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k}}{k \cdot k} \]

   10. Applied rewrites 63.7%

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k}}{k \cdot k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2200:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{\frac{l\_m}{k} \cdot \frac{l\_m}{k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{l\_m \cdot l\_m}{t}}{k \cdot k} \cdot \frac{2}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 5e-20)
   (* (/ 2.0 t) (/ (* (/ l_m k) (/ l_m k)) (* k k)))
   (* (/ (/ (* l_m l_m) t) (* k k)) (/ 2.0 (* k k)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 5e-20) {
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	} else {
		tmp = (((l_m * l_m) / t) / (k * k)) * (2.0 / (k * k));
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l_m <= 5d-20) then
        tmp = (2.0d0 / t) * (((l_m / k) * (l_m / k)) / (k * k))
    else
        tmp = (((l_m * l_m) / t) / (k * k)) * (2.0d0 / (k * k))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 5e-20) {
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	} else {
		tmp = (((l_m * l_m) / t) / (k * k)) * (2.0 / (k * k));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if l_m <= 5e-20:
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k))
	else:
		tmp = (((l_m * l_m) / t) / (k * k)) * (2.0 / (k * k))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 5e-20)
		tmp = Float64(Float64(2.0 / t) * Float64(Float64(Float64(l_m / k) * Float64(l_m / k)) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) / t) / Float64(k * k)) * Float64(2.0 / Float64(k * k)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (l_m <= 5e-20)
		tmp = (2.0 / t) * (((l_m / k) * (l_m / k)) / (k * k));
	else
		tmp = (((l_m * l_m) / t) / (k * k)) * (2.0 / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 5e-20], N[(N[(2.0 / t), $MachinePrecision] * N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999999e-20

   1. Initial program 31.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 72.2

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

      16. associate-*r/ N/A

          \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

   7. Applied rewrites 71.8%

      \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.5%

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{k \cdot k} \]

   11. Taylor expanded in k around 0

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{k} \cdot k} \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{k \cdot k} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{k \cdot k} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

       6. lift-/.f64 78.3

          \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot k} \]

   13. Applied rewrites 78.3%

       \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{k} \cdot k} \]

   if 4.9999999999999999e-20 < l

   1. Initial program 41.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-*.f64 67.8

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{2 \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{{k}^{4}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{\ell \cdot \ell}{t}}{{k}^{4}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{\ell \cdot \ell}{t}}{{k}^{4}} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      9. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t} \cdot 2}{\color{blue}{{k}^{2}} \cdot {k}^{2}} \]

      11. times-frac N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot \color{blue}{\frac{2}{{k}^{2}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot \color{blue}{\frac{2}{{k}^{2}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot \frac{\color{blue}{2}}{{k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      17. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k} \cdot \frac{2}{{k}^{2}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k} \cdot \frac{2}{{k}^{2}} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2}}} \]

      20. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k} \cdot \frac{2}{k \cdot \color{blue}{k}} \]

      21. lift-*.f64 70.7

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k} \cdot \frac{2}{k \cdot \color{blue}{k}} \]

   7. Applied rewrites 70.7%

      \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k} \cdot \color{blue}{\frac{2}{k \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 71.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{l\_m}{k} \cdot \frac{l\_m}{k}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ 2.0 (* (* k k) t)) (* (/ l_m k) (/ l_m k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k));
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 / ((k * k) * t)) * ((l_m / k) * (l_m / k))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l_m / k) * Float64(l_m / k)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 / ((k * k) * t)) * ((l_m / k) * (l_m / k));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 75.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
7. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   6. lower-/.f64 75.0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

8. Applied rewrites 75.0%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

9. Final simplification 75.0%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]
10. Add Preprocessing

Alternative 18: 64.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{t} \cdot \frac{\frac{l\_m \cdot l\_m}{k \cdot k}}{k \cdot k} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ 2.0 t) (/ (/ (* l_m l_m) (* k k)) (* k k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 / t) * (((l_m * l_m) / (k * k)) / (k * k));
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 / t) * (((l_m * l_m) / (k * k)) / (k * k))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (2.0 / t) * (((l_m * l_m) / (k * k)) / (k * k));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 / t) * (((l_m * l_m) / (k * k)) / (k * k))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 / t) * Float64(Float64(Float64(l_m * l_m) / Float64(k * k)) / Float64(k * k)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 / t) * (((l_m * l_m) / (k * k)) / (k * k));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 / t), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 75.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. associate-*l/ N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2} \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{t \cdot \color{blue}{{k}^{2}}} \]

   14. times-frac N/A

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}{{k}^{2}}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{2 \cdot 1}{t} \cdot \frac{\frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}}}{{k}^{2}} \]

   16. associate-*r/ N/A

       \[\leadsto \left(2 \cdot \frac{1}{t}\right) \cdot \frac{\color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}}}{{k}^{2}} \]

7. Applied rewrites 74.9%

   \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}}{k \cdot k}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{t} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{k} \cdot k} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{k \cdot k} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{k \cdot k} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{k \cdot k} \]

   4. pow2 N/A

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{k \cdot k} \]

   5. lift-*.f64 64.6

      \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{k \cdot k} \]

10. Applied rewrites 64.6%

    \[\leadsto \frac{2}{t} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{k} \cdot k} \]
11. Add Preprocessing

Alternative 19: 60.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{l\_m \cdot l\_m}{t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ 2.0 (* (* k k) (* k k))) (/ (* l_m l_m) t)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 / ((k * k) * (k * k))) * ((l_m * l_m) / t);
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 / ((k * k) * (k * k))) * ((l_m * l_m) / t)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (2.0 / ((k * k) * (k * k))) * ((l_m * l_m) / t);
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 / ((k * k) * (k * k))) * ((l_m * l_m) / t)

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(Float64(l_m * l_m) / t))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 / ((k * k) * (k * k))) * ((l_m * l_m) / t);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. times-frac N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   7. pow2 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-*.f64 60.1

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

5. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2}{{k}^{\left(2 + 2\right)}} \cdot \frac{\ell \cdot \ell}{t} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-*.f64 60.1

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t} \]

7. Applied rewrites 60.1%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]
8. Add Preprocessing

Alternative 20: 20.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{-0.11666666666666667 \cdot \left(l\_m \cdot l\_m\right)}{t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ (* -0.11666666666666667 (* l_m l_m)) t))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (-0.11666666666666667 * (l_m * l_m)) / t;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = ((-0.11666666666666667d0) * (l_m * l_m)) / t
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (-0.11666666666666667 * (l_m * l_m)) / t;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (-0.11666666666666667 * (l_m * l_m)) / t

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(-0.11666666666666667 * Float64(l_m * l_m)) / t)
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (-0.11666666666666667 * (l_m * l_m)) / t;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(-0.11666666666666667 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 29.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 16.7

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 16.7%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{t} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot \left(\ell \cdot \ell\right)}{t} \]

   9. lift-*.f64 16.7

      \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]

10. Applied rewrites 16.7%

    \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]
11. Add Preprocessing

Alternative 21: 20.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ -0.11666666666666667 \cdot \frac{l\_m \cdot l\_m}{t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* -0.11666666666666667 (/ (* l_m l_m) t)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return -0.11666666666666667 * ((l_m * l_m) / t);
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * ((l_m * l_m) / t)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return -0.11666666666666667 * ((l_m * l_m) / t);
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return -0.11666666666666667 * ((l_m * l_m) / t)

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(-0.11666666666666667 * Float64(Float64(l_m * l_m) / t))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = -0.11666666666666667 * ((l_m * l_m) / t);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(-0.11666666666666667 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 29.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 16.7

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 16.7%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Add Preprocessing

Alternative 22: 18.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ -0.11666666666666667 \cdot \left(l\_m \cdot \frac{l\_m}{t}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* -0.11666666666666667 (* l_m (/ l_m t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return -0.11666666666666667 * (l_m * (l_m / t));
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * (l_m * (l_m / t))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return -0.11666666666666667 * (l_m * (l_m / t));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return -0.11666666666666667 * (l_m * (l_m / t))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(-0.11666666666666667 * Float64(l_m * Float64(l_m / t)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = -0.11666666666666667 * (l_m * (l_m / t));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(-0.11666666666666667 * N[(l$95$m * N[(l$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 29.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 16.7

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 16.7%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 16.0

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

10. Applied rewrites 16.0%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025050 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  :pre (TRUE)
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))