Toniolo and Linder, Equation (10-)

Percentage Accurate: 37.2% → 94.8%

Time: 11.0s

Alternatives: 23

Speedup: 9.6×

• 40.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{\sin k\_m}^{2}} \cdot \frac{2}{k\_m}\right) \cdot \frac{\frac{\cos k\_m \cdot \ell}{t}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.8e-140)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (*
    (* (/ l (pow (sin k_m) 2.0)) (/ 2.0 k_m))
    (/ (/ (* (cos k_m) l) t) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.8e-140) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = ((l / pow(sin(k_m), 2.0)) * (2.0 / k_m)) * (((cos(k_m) * l) / t) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.8d-140) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else
        tmp = ((l / (sin(k_m) ** 2.0d0)) * (2.0d0 / k_m)) * (((cos(k_m) * l) / t) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.8e-140) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = ((l / Math.pow(Math.sin(k_m), 2.0)) * (2.0 / k_m)) * (((Math.cos(k_m) * l) / t) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.8e-140:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	else:
		tmp = ((l / math.pow(math.sin(k_m), 2.0)) * (2.0 / k_m)) * (((math.cos(k_m) * l) / t) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.8e-140)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(l / (sin(k_m) ^ 2.0)) * Float64(2.0 / k_m)) * Float64(Float64(Float64(cos(k_m) * l) / t) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.8e-140)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	else
		tmp = ((l / (sin(k_m) ^ 2.0)) * (2.0 / k_m)) * (((cos(k_m) * l) / t) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.8e-140], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.80000000000000038e-140

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.4%

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

   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

   8. Applied rewrites 76.6%

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

   10. Applied rewrites 80.9%

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

   12. Applied rewrites 80.9%

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

   if 7.80000000000000038e-140 < k

   1. Initial program 32.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.0%

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

   10. Applied rewrites 90.3%

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

   12. Applied rewrites 97.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{2}{k}\right) \cdot \frac{\frac{\cos k \cdot \ell}{t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 3.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 9.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\ell}{t\_1} \cdot \left(\left(2 \cdot \frac{\ell}{k\_m \cdot k\_m}\right) \cdot \frac{\cos k\_m}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos k\_m \cdot \ell\right) \cdot 2}{k\_m \cdot t} \cdot \frac{\ell}{k\_m}}{t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= k_m 3.7e-140)
     (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
     (if (<= k_m 9.2e+144)
       (* (/ l t_1) (* (* 2.0 (/ l (* k_m k_m))) (/ (cos k_m) t)))
       (/ (* (/ (* (* (cos k_m) l) 2.0) (* k_m t)) (/ l k_m)) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 3.7e-140) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 9.2e+144) {
		tmp = (l / t_1) * ((2.0 * (l / (k_m * k_m))) * (cos(k_m) / t));
	} else {
		tmp = ((((cos(k_m) * l) * 2.0) / (k_m * t)) * (l / k_m)) / t_1;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    if (k_m <= 3.7d-140) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else if (k_m <= 9.2d+144) then
        tmp = (l / t_1) * ((2.0d0 * (l / (k_m * k_m))) * (cos(k_m) / t))
    else
        tmp = ((((cos(k_m) * l) * 2.0d0) / (k_m * t)) * (l / k_m)) / t_1
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 3.7e-140) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 9.2e+144) {
		tmp = (l / t_1) * ((2.0 * (l / (k_m * k_m))) * (Math.cos(k_m) / t));
	} else {
		tmp = ((((Math.cos(k_m) * l) * 2.0) / (k_m * t)) * (l / k_m)) / t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if k_m <= 3.7e-140:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	elif k_m <= 9.2e+144:
		tmp = (l / t_1) * ((2.0 * (l / (k_m * k_m))) * (math.cos(k_m) / t))
	else:
		tmp = ((((math.cos(k_m) * l) * 2.0) / (k_m * t)) * (l / k_m)) / t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 3.7e-140)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 9.2e+144)
		tmp = Float64(Float64(l / t_1) * Float64(Float64(2.0 * Float64(l / Float64(k_m * k_m))) * Float64(cos(k_m) / t)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) * l) * 2.0) / Float64(k_m * t)) * Float64(l / k_m)) / t_1);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 3.7e-140)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	elseif (k_m <= 9.2e+144)
		tmp = (l / t_1) * ((2.0 * (l / (k_m * k_m))) * (cos(k_m) / t));
	else
		tmp = ((((cos(k_m) * l) * 2.0) / (k_m * t)) * (l / k_m)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 3.7e-140], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 9.2e+144], N[(N[(l / t$95$1), $MachinePrecision] * N[(N[(2.0 * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.69999999999999977e-140

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.4%

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

   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

   8. Applied rewrites 76.6%

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

   10. Applied rewrites 80.9%

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

   12. Applied rewrites 80.9%

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

   if 3.69999999999999977e-140 < k < 9.2000000000000006e144

   1. Initial program 31.6%

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 98.5%

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

   if 9.2000000000000006e144 < k

   1. Initial program 33.4%

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.2%

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

   10. Applied rewrites 52.9%

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

   12. Applied rewrites 83.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\ell}{{\sin k}^{2}} \cdot \left(\left(2 \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k \cdot t} \cdot \frac{\ell}{k}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \left(\cos k\_m \cdot \ell\right) \cdot 2\\ t_2 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 8.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{t\_1}{t} \cdot \frac{\ell}{k\_m \cdot k\_m}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{k\_m \cdot t} \cdot \frac{\ell}{k\_m}}{t\_2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (* (cos k_m) l) 2.0)) (t_2 (pow (sin k_m) 2.0)))
   (if (<= k_m 8.5e-132)
     (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
     (if (<= k_m 8.5e+144)
       (/ (* (/ t_1 t) (/ l (* k_m k_m))) t_2)
       (/ (* (/ t_1 (* k_m t)) (/ l k_m)) t_2)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (cos(k_m) * l) * 2.0;
	double t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.5e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 8.5e+144) {
		tmp = ((t_1 / t) * (l / (k_m * k_m))) / t_2;
	} else {
		tmp = ((t_1 / (k_m * t)) * (l / k_m)) / t_2;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (cos(k_m) * l) * 2.0d0
    t_2 = sin(k_m) ** 2.0d0
    if (k_m <= 8.5d-132) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else if (k_m <= 8.5d+144) then
        tmp = ((t_1 / t) * (l / (k_m * k_m))) / t_2
    else
        tmp = ((t_1 / (k_m * t)) * (l / k_m)) / t_2
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (Math.cos(k_m) * l) * 2.0;
	double t_2 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.5e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 8.5e+144) {
		tmp = ((t_1 / t) * (l / (k_m * k_m))) / t_2;
	} else {
		tmp = ((t_1 / (k_m * t)) * (l / k_m)) / t_2;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (math.cos(k_m) * l) * 2.0
	t_2 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if k_m <= 8.5e-132:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	elif k_m <= 8.5e+144:
		tmp = ((t_1 / t) * (l / (k_m * k_m))) / t_2
	else:
		tmp = ((t_1 / (k_m * t)) * (l / k_m)) / t_2
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(cos(k_m) * l) * 2.0)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 8.5e-132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 8.5e+144)
		tmp = Float64(Float64(Float64(t_1 / t) * Float64(l / Float64(k_m * k_m))) / t_2);
	else
		tmp = Float64(Float64(Float64(t_1 / Float64(k_m * t)) * Float64(l / k_m)) / t_2);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (cos(k_m) * l) * 2.0;
	t_2 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 8.5e-132)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	elseif (k_m <= 8.5e+144)
		tmp = ((t_1 / t) * (l / (k_m * k_m))) / t_2;
	else
		tmp = ((t_1 / (k_m * t)) * (l / k_m)) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 8.5e-132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.5e+144], N[(N[(N[(t$95$1 / t), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(t$95$1 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.49999999999999988e-132

   1. Initial program 36.3%

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

   5. Applied rewrites 73.9%

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

   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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 81.2%

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

   12. Applied rewrites 81.3%

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

   if 8.49999999999999988e-132 < k < 8.4999999999999998e144

   1. Initial program 27.9%

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 87.7%

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

   10. Applied rewrites 95.5%

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

   12. Applied rewrites 97.6%

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

   if 8.4999999999999998e144 < k

   1. Initial program 33.4%

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.2%

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

   10. Applied rewrites 52.9%

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

   12. Applied rewrites 83.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\left(\cos k \cdot \ell\right) \cdot 2}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k \cdot t} \cdot \frac{\ell}{k}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 8.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\left(\cos k\_m \cdot \ell\right) \cdot 2}{t} \cdot \frac{\ell}{k\_m \cdot k\_m}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{k\_m} \cdot \ell\right) \cdot \cos k\_m\right) \cdot \frac{\ell}{t\_1 \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= k_m 8.5e-132)
     (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
     (if (<= k_m 8.4e+144)
       (/ (* (/ (* (* (cos k_m) l) 2.0) t) (/ l (* k_m k_m))) t_1)
       (* (* (* (/ 2.0 k_m) l) (cos k_m)) (/ l (* t_1 (* k_m t))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.5e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 8.4e+144) {
		tmp = ((((cos(k_m) * l) * 2.0) / t) * (l / (k_m * k_m))) / t_1;
	} else {
		tmp = (((2.0 / k_m) * l) * cos(k_m)) * (l / (t_1 * (k_m * t)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    if (k_m <= 8.5d-132) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else if (k_m <= 8.4d+144) then
        tmp = ((((cos(k_m) * l) * 2.0d0) / t) * (l / (k_m * k_m))) / t_1
    else
        tmp = (((2.0d0 / k_m) * l) * cos(k_m)) * (l / (t_1 * (k_m * t)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.5e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 8.4e+144) {
		tmp = ((((Math.cos(k_m) * l) * 2.0) / t) * (l / (k_m * k_m))) / t_1;
	} else {
		tmp = (((2.0 / k_m) * l) * Math.cos(k_m)) * (l / (t_1 * (k_m * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if k_m <= 8.5e-132:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	elif k_m <= 8.4e+144:
		tmp = ((((math.cos(k_m) * l) * 2.0) / t) * (l / (k_m * k_m))) / t_1
	else:
		tmp = (((2.0 / k_m) * l) * math.cos(k_m)) * (l / (t_1 * (k_m * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 8.5e-132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 8.4e+144)
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) * l) * 2.0) / t) * Float64(l / Float64(k_m * k_m))) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * l) * cos(k_m)) * Float64(l / Float64(t_1 * Float64(k_m * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 8.5e-132)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	elseif (k_m <= 8.4e+144)
		tmp = ((((cos(k_m) * l) * 2.0) / t) * (l / (k_m * k_m))) / t_1;
	else
		tmp = (((2.0 / k_m) * l) * cos(k_m)) * (l / (t_1 * (k_m * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 8.5e-132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.4e+144], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * l), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$1 * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.49999999999999988e-132

   1. Initial program 36.3%

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

   5. Applied rewrites 73.9%

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

   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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 81.2%

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

   12. Applied rewrites 81.3%

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

   if 8.49999999999999988e-132 < k < 8.39999999999999985e144

   1. Initial program 27.9%

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 87.7%

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

   10. Applied rewrites 95.5%

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

   12. Applied rewrites 97.6%

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

   if 8.39999999999999985e144 < k

   1. Initial program 33.4%

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.2%

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

   10. Applied rewrites 81.0%

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

   12. Applied rewrites 83.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 8.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\left(\cos k \cdot \ell\right) \cdot 2}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{k} \cdot \ell\right) \cdot \cos k\right) \cdot \frac{\ell}{{\sin k}^{2} \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\ell \cdot \mathsf{fma}\left(-k\_m, k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot \frac{\ell}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{k\_m} \cdot \ell\right) \cdot \cos k\_m\right) \cdot \frac{\ell}{t\_1 \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= k_m 7.8e-140)
     (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
     (if (<= k_m 3.6e-5)
       (* (/ (/ (* l (fma (- k_m) k_m 2.0)) t) (* k_m k_m)) (/ l t_1))
       (* (* (* (/ 2.0 k_m) l) (cos k_m)) (/ l (* t_1 (* k_m t))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 7.8e-140) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 3.6e-5) {
		tmp = (((l * fma(-k_m, k_m, 2.0)) / t) / (k_m * k_m)) * (l / t_1);
	} else {
		tmp = (((2.0 / k_m) * l) * cos(k_m)) * (l / (t_1 * (k_m * t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 7.8e-140)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 3.6e-5)
		tmp = Float64(Float64(Float64(Float64(l * fma(Float64(-k_m), k_m, 2.0)) / t) / Float64(k_m * k_m)) * Float64(l / t_1));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * l) * cos(k_m)) * Float64(l / Float64(t_1 * Float64(k_m * t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 7.8e-140], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.6e-5], N[(N[(N[(N[(l * N[((-k$95$m) * k$95$m + 2.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * l), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$1 * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.80000000000000038e-140

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.4%

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

   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

   8. Applied rewrites 76.6%

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

   10. Applied rewrites 80.9%

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

   12. Applied rewrites 80.9%

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

   if 7.80000000000000038e-140 < k < 3.60000000000000009e-5

   1. Initial program 36.0%

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.0%

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

   10. Applied rewrites 97.3%

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

   11. Taylor expanded in k around 0

       \[\leadsto \frac{-1 \cdot \frac{{k}^{2} \cdot \ell}{t} + 2 \cdot \frac{\ell}{t}}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{\sin k}^{2}} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.7%

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

   if 3.60000000000000009e-5 < k

   1. Initial program 30.3%

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.8%

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

   10. Applied rewrites 86.7%

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

   12. Applied rewrites 89.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\ell \cdot \mathsf{fma}\left(-k, k, 2\right)}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{k} \cdot \ell\right) \cdot \cos k\right) \cdot \frac{\ell}{{\sin k}^{2} \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)}{k\_m \cdot t} \cdot \frac{\ell}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.3e-132)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (*
    (/ (* (/ 2.0 k_m) (* (cos k_m) l)) (* k_m t))
    (/ l (pow (sin k_m) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (((2.0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (l / pow(sin(k_m), 2.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.3d-132) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else
        tmp = (((2.0d0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (l / (sin(k_m) ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (((2.0 / k_m) * (Math.cos(k_m) * l)) / (k_m * t)) * (l / Math.pow(Math.sin(k_m), 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.3e-132:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	else:
		tmp = (((2.0 / k_m) * (math.cos(k_m) * l)) / (k_m * t)) * (l / math.pow(math.sin(k_m), 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e-132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) / Float64(k_m * t)) * Float64(l / (sin(k_m) ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.3e-132)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	else
		tmp = (((2.0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (l / (sin(k_m) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.3e-132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-132

   1. Initial program 36.3%

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

   5. Applied rewrites 73.9%

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

   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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 81.2%

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

   12. Applied rewrites 81.3%

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

   if 1.3e-132 < k

   1. Initial program 30.0%

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 80.4%

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

   10. Applied rewrites 90.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\cos k \cdot \ell\right)}{k \cdot t} \cdot \frac{\ell}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 0.0028:\\ \;\;\;\;\frac{\frac{\ell \cdot \mathsf{fma}\left(-k\_m, k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot \frac{\ell}{{\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)}{k\_m \cdot t} \cdot \frac{\ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.8e-140)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (if (<= k_m 0.0028)
     (*
      (/ (/ (* l (fma (- k_m) k_m 2.0)) t) (* k_m k_m))
      (/ l (pow (sin k_m) 2.0)))
     (*
      (/ (* (/ 2.0 k_m) (* (cos k_m) l)) (* k_m t))
      (/ l (- 0.5 (* 0.5 (cos (* 2.0 k_m)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.8e-140) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 0.0028) {
		tmp = (((l * fma(-k_m, k_m, 2.0)) / t) / (k_m * k_m)) * (l / pow(sin(k_m), 2.0));
	} else {
		tmp = (((2.0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (l / (0.5 - (0.5 * cos((2.0 * k_m)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.8e-140)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 0.0028)
		tmp = Float64(Float64(Float64(Float64(l * fma(Float64(-k_m), k_m, 2.0)) / t) / Float64(k_m * k_m)) * Float64(l / (sin(k_m) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) / Float64(k_m * t)) * Float64(l / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.8e-140], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.0028], N[(N[(N[(N[(l * N[((-k$95$m) * k$95$m + 2.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.80000000000000038e-140

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.4%

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

   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

   8. Applied rewrites 76.6%

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

   10. Applied rewrites 80.9%

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

   12. Applied rewrites 80.9%

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

   if 7.80000000000000038e-140 < k < 0.00279999999999999997

   1. Initial program 36.0%

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.0%

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

   10. Applied rewrites 97.3%

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

   11. Taylor expanded in k around 0

       \[\leadsto \frac{-1 \cdot \frac{{k}^{2} \cdot \ell}{t} + 2 \cdot \frac{\ell}{t}}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{\sin k}^{2}} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.7%

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

   if 0.00279999999999999997 < k

   1. Initial program 30.3%

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.8%

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

   10. Applied rewrites 86.7%

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

   12. Applied rewrites 86.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 0.0028:\\ \;\;\;\;\frac{\frac{\ell \cdot \mathsf{fma}\left(-k, k, 2\right)}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\cos k \cdot \ell\right)}{k \cdot t} \cdot \frac{\ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{{\sin k\_m}^{2}}\\ \mathbf{if}\;k\_m \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 0.028:\\ \;\;\;\;\frac{\frac{\ell \cdot \mathsf{fma}\left(-k\_m, k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot t\_1\\ \mathbf{elif}\;k\_m \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot \ell}{k\_m \cdot t} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (pow (sin k_m) 2.0))))
   (if (<= k_m 7.8e-140)
     (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
     (if (<= k_m 0.028)
       (* (/ (/ (* l (fma (- k_m) k_m 2.0)) t) (* k_m k_m)) t_1)
       (if (<= k_m 7.2e+144)
         (/
          (* 2.0 (* (* (cos k_m) l) l))
          (* (* (* k_m k_m) t) (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
         (* (/ (* (/ 2.0 k_m) l) (* k_m t)) t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 7.8e-140) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 0.028) {
		tmp = (((l * fma(-k_m, k_m, 2.0)) / t) / (k_m * k_m)) * t_1;
	} else if (k_m <= 7.2e+144) {
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	} else {
		tmp = (((2.0 / k_m) * l) / (k_m * t)) * t_1;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / (sin(k_m) ^ 2.0))
	tmp = 0.0
	if (k_m <= 7.8e-140)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 0.028)
		tmp = Float64(Float64(Float64(Float64(l * fma(Float64(-k_m), k_m, 2.0)) / t) / Float64(k_m * k_m)) * t_1);
	elseif (k_m <= 7.2e+144)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l) * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * l) / Float64(k_m * t)) * t_1);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 7.8e-140], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.028], N[(N[(N[(N[(l * N[((-k$95$m) * k$95$m + 2.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[k$95$m, 7.2e+144], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 7.80000000000000038e-140

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.4%

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

   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

   8. Applied rewrites 76.6%

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

   10. Applied rewrites 80.9%

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

   12. Applied rewrites 80.9%

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

   if 7.80000000000000038e-140 < k < 0.0280000000000000006

   1. Initial program 36.0%

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.0%

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

   10. Applied rewrites 97.3%

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

   11. Taylor expanded in k around 0

       \[\leadsto \frac{-1 \cdot \frac{{k}^{2} \cdot \ell}{t} + 2 \cdot \frac{\ell}{t}}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{\sin k}^{2}} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.7%

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

   if 0.0280000000000000006 < k < 7.1999999999999995e144

   1. Initial program 27.3%

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 73.9%

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

   9. Applied rewrites 73.9%

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

   if 7.1999999999999995e144 < k

   1. Initial program 32.5%

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.3%

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

   10. Applied rewrites 78.8%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 55.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 0.028:\\ \;\;\;\;\frac{\frac{\ell \cdot \mathsf{fma}\left(-k, k, 2\right)}{t}}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \ell}{k \cdot t} \cdot \frac{\ell}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 0.028:\\ \;\;\;\;\left(\frac{\frac{2}{k\_m}}{k\_m \cdot t} \cdot t\_1\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}\\ \mathbf{elif}\;k\_m \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot \ell}{k\_m \cdot t} \cdot \frac{\ell}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 0.028)
     (*
      (* (/ (/ 2.0 k_m) (* k_m t)) t_1)
      (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m))
     (if (<= k_m 7.2e+144)
       (/
        (* 2.0 (* t_1 l))
        (* (* (* k_m k_m) t) (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
       (* (/ (* (/ 2.0 k_m) l) (* k_m t)) (/ l (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 0.028) {
		tmp = (((2.0 / k_m) / (k_m * t)) * t_1) * ((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m);
	} else if (k_m <= 7.2e+144) {
		tmp = (2.0 * (t_1 * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	} else {
		tmp = (((2.0 / k_m) * l) / (k_m * t)) * (l / pow(sin(k_m), 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 0.028)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / Float64(k_m * t)) * t_1) * Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m));
	elseif (k_m <= 7.2e+144)
		tmp = Float64(Float64(2.0 * Float64(t_1 * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * l) / Float64(k_m * t)) * Float64(l / (sin(k_m) ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.028], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7.2e+144], N[(N[(2.0 * N[(t$95$1 * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.0280000000000000006

   1. Initial program 35.3%

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 79.6%

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

   if 0.0280000000000000006 < k < 7.1999999999999995e144

   1. Initial program 27.3%

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 73.9%

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

   9. Applied rewrites 73.9%

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

   if 7.1999999999999995e144 < k

   1. Initial program 32.5%

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.3%

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

   10. Applied rewrites 78.8%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 55.6%

       \[\leadsto \frac{\frac{2}{k} \cdot \ell}{k \cdot t} \cdot \frac{\ell}{{\sin k}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.028:\\ \;\;\;\;\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k \cdot k\right) \cdot \ell, \ell\right)}{k}}{k}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \ell}{k \cdot t} \cdot \frac{\ell}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.3 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{\frac{2}{k\_m}}{k\_m \cdot t} \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot \ell}{k\_m \cdot t} \cdot \frac{\ell}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.3e+132)
   (*
    (* (/ (/ 2.0 k_m) (* k_m t)) (* (cos k_m) l))
    (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m))
   (* (/ (* (/ 2.0 k_m) l) (* k_m t)) (/ l (pow (sin k_m) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.3e+132) {
		tmp = (((2.0 / k_m) / (k_m * t)) * (cos(k_m) * l)) * ((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m);
	} else {
		tmp = (((2.0 / k_m) * l) / (k_m * t)) * (l / pow(sin(k_m), 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.3e+132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / Float64(k_m * t)) * Float64(cos(k_m) * l)) * Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * l) / Float64(k_m * t)) * Float64(l / (sin(k_m) ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.3e+132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.3e132

   1. Initial program 34.3%

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 75.9%

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

   if 5.3e132 < k

   1. Initial program 32.5%

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.3%

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

   10. Applied rewrites 78.8%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 55.6%

       \[\leadsto \frac{\frac{2}{k} \cdot \ell}{k \cdot t} \cdot \frac{\ell}{{\sin k}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.3 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k \cdot k\right) \cdot \ell, \ell\right)}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \ell}{k \cdot t} \cdot \frac{\ell}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.3 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{\frac{2}{k\_m}}{k\_m \cdot t} \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot 2\right) \cdot \frac{\ell}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.3e+132)
   (*
    (* (/ (/ 2.0 k_m) (* k_m t)) (* (cos k_m) l))
    (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m))
   (* (* (/ l (* (* k_m t) k_m)) 2.0) (/ l (pow (sin k_m) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.3e+132) {
		tmp = (((2.0 / k_m) / (k_m * t)) * (cos(k_m) * l)) * ((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m);
	} else {
		tmp = ((l / ((k_m * t) * k_m)) * 2.0) * (l / pow(sin(k_m), 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.3e+132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / Float64(k_m * t)) * Float64(cos(k_m) * l)) * Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m));
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(k_m * t) * k_m)) * 2.0) * Float64(l / (sin(k_m) ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.3e+132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.3e132

   1. Initial program 34.3%

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 75.9%

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

   if 5.3e132 < k

   1. Initial program 32.5%

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 54.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.3 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k \cdot k\right) \cdot \ell, \ell\right)}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(k \cdot t\right) \cdot k} \cdot 2\right) \cdot \frac{\ell}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)}{k\_m \cdot t} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.3e-132)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (if (<= k_m 1.65e+133)
     (*
      (/ (* (/ 2.0 k_m) (* (cos k_m) l)) (* k_m t))
      (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) (* k_m k_m)))
     (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 1.65e+133) {
		tmp = (((2.0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (fma(0.3333333333333333, ((k_m * k_m) * l), l) / (k_m * k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e-132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 1.65e+133)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) / Float64(k_m * t)) * Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.3e-132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.65e+133], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.3e-132

   1. Initial program 36.3%

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

   5. Applied rewrites 73.9%

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

   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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 81.2%

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

   12. Applied rewrites 81.3%

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

   if 1.3e-132 < k < 1.65e133

   1. Initial program 28.4%

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.3%

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

   10. Applied rewrites 97.1%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 74.5%

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

   if 1.65e133 < k

   1. Initial program 32.5%

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

   5. Applied rewrites 47.3%

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

   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

   8. Applied rewrites 51.4%

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

   10. Applied rewrites 51.4%

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

   12. Applied rewrites 53.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\cos k \cdot \ell\right)}{k \cdot t} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, \left(k \cdot k\right) \cdot \ell, \ell\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{\frac{2}{k\_m}}{k\_m \cdot t} \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8e+132)
   (*
    (* (/ (/ 2.0 k_m) (* k_m t)) (* (cos k_m) l))
    (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m))
   (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8e+132) {
		tmp = (((2.0 / k_m) / (k_m * t)) * (cos(k_m) * l)) * ((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m);
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8e+132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / Float64(k_m * t)) * Float64(cos(k_m) * l)) * Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8e+132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.99999999999999993e132

   1. Initial program 34.3%

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 75.9%

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

   if 7.99999999999999993e132 < k

   1. Initial program 32.5%

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

   5. Applied rewrites 47.3%

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

   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

   8. Applied rewrites 51.4%

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

   10. Applied rewrites 51.4%

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

   12. Applied rewrites 53.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k \cdot k\right) \cdot \ell, \ell\right)}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 3.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)}{k\_m \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.3e-132)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (if (<= k_m 3.9e+99)
     (* (/ (* (/ 2.0 k_m) (* (cos k_m) l)) (* k_m t)) (/ l (* k_m k_m)))
     (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 3.9e+99) {
		tmp = (((2.0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (l / (k_m * k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.3d-132) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else if (k_m <= 3.9d+99) then
        tmp = (((2.0d0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (l / (k_m * k_m))
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-132) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 3.9e+99) {
		tmp = (((2.0 / k_m) * (Math.cos(k_m) * l)) / (k_m * t)) * (l / (k_m * k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.3e-132:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	elif k_m <= 3.9e+99:
		tmp = (((2.0 / k_m) * (math.cos(k_m) * l)) / (k_m * t)) * (l / (k_m * k_m))
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e-132)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 3.9e+99)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) / Float64(k_m * t)) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.3e-132)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	elseif (k_m <= 3.9e+99)
		tmp = (((2.0 / k_m) * (cos(k_m) * l)) / (k_m * t)) * (l / (k_m * k_m));
	else
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.3e-132], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.9e+99], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.3e-132

   1. Initial program 36.3%

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

   5. Applied rewrites 73.9%

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

   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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 81.2%

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

   12. Applied rewrites 81.3%

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

   if 1.3e-132 < k < 3.89999999999999995e99

   1. Initial program 25.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 73.7%

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

   if 3.89999999999999995e99 < k

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 51.6%

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

   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

   8. Applied rewrites 48.1%

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

   10. Applied rewrites 48.1%

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

   12. Applied rewrites 54.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\cos k \cdot \ell\right)}{k \cdot t} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{elif}\;k\_m \leq 0.4:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{k\_m \cdot k\_m}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.5e-79)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (if (<= k_m 0.4)
     (/ 2.0 (* (/ (pow k_m 4.0) l) (/ t l)))
     (/ 2.0 (* (* (* k_m k_m) t) (/ (* k_m k_m) (* (* (cos k_m) l) l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else if (k_m <= 0.4) {
		tmp = 2.0 / ((pow(k_m, 4.0) / l) * (t / l));
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / ((cos(k_m) * l) * l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.5d-79) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else if (k_m <= 0.4d0) then
        tmp = 2.0d0 / (((k_m ** 4.0d0) / l) * (t / l))
    else
        tmp = 2.0d0 / (((k_m * k_m) * t) * ((k_m * k_m) / ((cos(k_m) * l) * l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else if (k_m <= 0.4) {
		tmp = 2.0 / ((Math.pow(k_m, 4.0) / l) * (t / l));
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / ((Math.cos(k_m) * l) * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.5e-79:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	elif k_m <= 0.4:
		tmp = 2.0 / ((math.pow(k_m, 4.0) / l) * (t / l))
	else:
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / ((math.cos(k_m) * l) * l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-79)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	elseif (k_m <= 0.4)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 4.0) / l) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m * k_m) / Float64(Float64(cos(k_m) * l) * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.5e-79)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	elseif (k_m <= 0.4)
		tmp = 2.0 / (((k_m ^ 4.0) / l) * (t / l));
	else
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / ((cos(k_m) * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.5e-79], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.4], N[(2.0 / N[(N[(N[Power[k$95$m, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5e-79

   1. Initial program 36.3%

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

   5. Applied rewrites 74.3%

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

   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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 81.1%

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

   12. Applied rewrites 82.3%

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

   if 1.5e-79 < k < 0.40000000000000002

   1. Initial program 23.0%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 99.5%

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

   if 0.40000000000000002 < k

   1. Initial program 30.3%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 46.6%

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

   10. Applied rewrites 46.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{elif}\;k \leq 0.4:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{elif}\;k\_m \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.5e-79)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (if (<= k_m 5e+41)
     (/ 2.0 (* (/ (pow k_m 4.0) l) (/ t l)))
     (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else if (k_m <= 5e+41) {
		tmp = 2.0 / ((pow(k_m, 4.0) / l) * (t / l));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.5d-79) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else if (k_m <= 5d+41) then
        tmp = 2.0d0 / (((k_m ** 4.0d0) / l) * (t / l))
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else if (k_m <= 5e+41) {
		tmp = 2.0 / ((Math.pow(k_m, 4.0) / l) * (t / l));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.5e-79:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	elif k_m <= 5e+41:
		tmp = 2.0 / ((math.pow(k_m, 4.0) / l) * (t / l))
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-79)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	elseif (k_m <= 5e+41)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 4.0) / l) * Float64(t / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.5e-79)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	elseif (k_m <= 5e+41)
		tmp = 2.0 / (((k_m ^ 4.0) / l) * (t / l));
	else
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.5e-79], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5e+41], N[(2.0 / N[(N[(N[Power[k$95$m, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5e-79

   1. Initial program 36.3%

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

   5. Applied rewrites 74.3%

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

   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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 81.1%

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

   12. Applied rewrites 82.3%

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

   if 1.5e-79 < k < 5.00000000000000022e41

   1. Initial program 24.9%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 73.5%

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

   if 5.00000000000000022e41 < k

   1. Initial program 30.5%

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

   5. Applied rewrites 55.1%

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

   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

   8. Applied rewrites 43.3%

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

   10. Applied rewrites 43.3%

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

   12. Applied rewrites 48.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 75.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.5e-79)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (if (<= k_m 5e+41)
     (/ 2.0 (* (/ (pow k_m 4.0) l) (/ t l)))
     (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 5e+41) {
		tmp = 2.0 / ((pow(k_m, 4.0) / l) * (t / l));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.5d-79) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else if (k_m <= 5d+41) then
        tmp = 2.0d0 / (((k_m ** 4.0d0) / l) * (t / l))
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 5e+41) {
		tmp = 2.0 / ((Math.pow(k_m, 4.0) / l) * (t / l));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.5e-79:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	elif k_m <= 5e+41:
		tmp = 2.0 / ((math.pow(k_m, 4.0) / l) * (t / l))
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-79)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 5e+41)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 4.0) / l) * Float64(t / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.5e-79)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	elseif (k_m <= 5e+41)
		tmp = 2.0 / (((k_m ^ 4.0) / l) * (t / l));
	else
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.5e-79], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5e+41], N[(2.0 / N[(N[(N[Power[k$95$m, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5e-79

   1. Initial program 36.3%

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

   5. Applied rewrites 74.3%

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

   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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 81.1%

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

   12. Applied rewrites 81.1%

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

   if 1.5e-79 < k < 5.00000000000000022e41

   1. Initial program 24.9%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 73.5%

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

   if 5.00000000000000022e41 < k

   1. Initial program 30.5%

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

   5. Applied rewrites 55.1%

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

   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

   8. Applied rewrites 43.3%

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

   10. Applied rewrites 43.3%

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

   12. Applied rewrites 48.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 10^{+40}:\\ \;\;\;\;\frac{\frac{\ell}{{k\_m}^{4}}}{t} \cdot \left(\ell + \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.5e-79)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (if (<= k_m 1e+40)
     (* (/ (/ l (pow k_m 4.0)) t) (+ l l))
     (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 1e+40) {
		tmp = ((l / pow(k_m, 4.0)) / t) * (l + l);
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.5d-79) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else if (k_m <= 1d+40) then
        tmp = ((l / (k_m ** 4.0d0)) / t) * (l + l)
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-79) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 1e+40) {
		tmp = ((l / Math.pow(k_m, 4.0)) / t) * (l + l);
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.5e-79:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	elif k_m <= 1e+40:
		tmp = ((l / math.pow(k_m, 4.0)) / t) * (l + l)
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-79)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 1e+40)
		tmp = Float64(Float64(Float64(l / (k_m ^ 4.0)) / t) * Float64(l + l));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.5e-79)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	elseif (k_m <= 1e+40)
		tmp = ((l / (k_m ^ 4.0)) / t) * (l + l);
	else
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.5e-79], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1e+40], N[(N[(N[(l / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5e-79

   1. Initial program 36.3%

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

   5. Applied rewrites 74.3%

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

   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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 81.1%

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

   12. Applied rewrites 81.1%

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

   if 1.5e-79 < k < 1.00000000000000003e40

   1. Initial program 24.9%

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 73.6%

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

   if 1.00000000000000003e40 < k

   1. Initial program 30.5%

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

   5. Applied rewrites 55.1%

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

   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

   8. Applied rewrites 43.3%

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

   10. Applied rewrites 43.3%

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

   12. Applied rewrites 48.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 10^{+40}:\\ \;\;\;\;\frac{\frac{\ell}{{k}^{4}}}{t} \cdot \left(\ell + \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 74.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2e-20)
   (* (/ (/ (/ 2.0 k_m) t) k_m) (* (/ l k_m) (/ l k_m)))
   (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-20) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2d-20) then
        tmp = (((2.0d0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-20) {
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2e-20:
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m))
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2e-20)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / t) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2e-20)
		tmp = (((2.0 / k_m) / t) / k_m) * ((l / k_m) * (l / k_m));
	else
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2e-20], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999989e-20

   1. Initial program 36.4%

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

   5. Applied rewrites 73.9%

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

   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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 80.9%

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

   12. Applied rewrites 80.9%

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

   if 1.99999999999999989e-20 < k

   1. Initial program 27.7%

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

   5. Applied rewrites 62.9%

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

   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

   8. Applied rewrites 46.4%

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

   10. Applied rewrites 46.4%

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

   12. Applied rewrites 50.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 74.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{2}{k\_m \cdot t}}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2e-20)
   (* (/ (/ 2.0 (* k_m t)) k_m) (* (/ l k_m) (/ l k_m)))
   (* (/ 2.0 (* k_m (* k_m t))) (/ (* (/ l k_m) l) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-20) {
		tmp = ((2.0 / (k_m * t)) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2d-20) then
        tmp = ((2.0d0 / (k_m * t)) / k_m) * ((l / k_m) * (l / k_m))
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-20) {
		tmp = ((2.0 / (k_m * t)) / k_m) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2e-20:
		tmp = ((2.0 / (k_m * t)) / k_m) * ((l / k_m) * (l / k_m))
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2e-20)
		tmp = Float64(Float64(Float64(2.0 / Float64(k_m * t)) / k_m) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2e-20)
		tmp = ((2.0 / (k_m * t)) / k_m) * ((l / k_m) * (l / k_m));
	else
		tmp = (2.0 / (k_m * (k_m * t))) * (((l / k_m) * l) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2e-20], N[(N[(N[(2.0 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999989e-20

   1. Initial program 36.4%

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

   5. Applied rewrites 73.9%

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

   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

   8. Applied rewrites 77.2%

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

   10. Applied rewrites 80.9%

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

   12. Applied rewrites 80.9%

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

   if 1.99999999999999989e-20 < k

   1. Initial program 27.7%

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

   5. Applied rewrites 62.9%

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

   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

   8. Applied rewrites 46.4%

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

   10. Applied rewrites 46.4%

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

   12. Applied rewrites 50.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{2}{k \cdot t}}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 73.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{2}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{if}\;\ell \leq 2 \cdot 10^{-122}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell}{k\_m} \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* k_m (* k_m t)))))
   (if (<= l 2e-122)
     (* t_1 (* (/ l k_m) (/ l k_m)))
     (* t_1 (/ (* (/ l k_m) l) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 2.0 / (k_m * (k_m * t));
	double tmp;
	if (l <= 2e-122) {
		tmp = t_1 * ((l / k_m) * (l / k_m));
	} else {
		tmp = t_1 * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (k_m * (k_m * t))
    if (l <= 2d-122) then
        tmp = t_1 * ((l / k_m) * (l / k_m))
    else
        tmp = t_1 * (((l / k_m) * l) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 2.0 / (k_m * (k_m * t));
	double tmp;
	if (l <= 2e-122) {
		tmp = t_1 * ((l / k_m) * (l / k_m));
	} else {
		tmp = t_1 * (((l / k_m) * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 2.0 / (k_m * (k_m * t))
	tmp = 0
	if l <= 2e-122:
		tmp = t_1 * ((l / k_m) * (l / k_m))
	else:
		tmp = t_1 * (((l / k_m) * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(2.0 / Float64(k_m * Float64(k_m * t)))
	tmp = 0.0
	if (l <= 2e-122)
		tmp = Float64(t_1 * Float64(Float64(l / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(l / k_m) * l) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 2.0 / (k_m * (k_m * t));
	tmp = 0.0;
	if (l <= 2e-122)
		tmp = t_1 * ((l / k_m) * (l / k_m));
	else
		tmp = t_1 * (((l / k_m) * l) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2e-122], N[(t$95$1 * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.00000000000000012e-122

   1. Initial program 33.4%

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

   5. Applied rewrites 69.5%

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

   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

   8. Applied rewrites 69.9%

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

   10. Applied rewrites 73.5%

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

   if 2.00000000000000012e-122 < l

   1. Initial program 35.5%

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

   5. Applied rewrites 73.8%

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

   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

   8. Applied rewrites 67.2%

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

   10. Applied rewrites 68.1%

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

   12. Applied rewrites 70.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-122}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 73.8% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 / (k_m * (k_m * t))) * ((l / k_m) * (l / k_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 70.9%

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

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

8. Applied rewrites 69.0%

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

10. Applied rewrites 71.7%

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

11. Final simplification 71.7%

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

Alternative 23: 72.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 / (k_m * (k_m * t))) * (l * (l / (k_m * k_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 70.9%

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

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

8. Applied rewrites 69.0%

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

10. Applied rewrites 71.7%

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

12. Applied rewrites 69.0%

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

13. Final simplification 69.0%

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

Reproduce

herbie shell --seed 2025019 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))