Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 95.0%

Time: 9.5s

Alternatives: 17

Speedup: 8.6×

• 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: 35.8% 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: 95.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{t}\right) \cdot \ell}{t\_1} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\right) \cdot 4}{\left(t\_1 \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

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 3.2e-85)
     (* (/ 2.0 (* k_m (* k_m t))) (pow (/ l k_m) 2.0))
     (if (<= k_m 1.5e+124)
       (* (/ (* (* (/ l (* k_m k_m)) (/ (cos k_m) t)) l) t_1) 2.0)
       (/ (* (* (* (cos k_m) (/ l k_m)) (/ l k_m)) 4.0) (* (* t_1 t) 2.0))))))

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 <= 3.2e-85) {
		tmp = (2.0 / (k_m * (k_m * t))) * pow((l / k_m), 2.0);
	} else if (k_m <= 1.5e+124) {
		tmp = ((((l / (k_m * k_m)) * (cos(k_m) / t)) * l) / t_1) * 2.0;
	} else {
		tmp = (((cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / ((t_1 * t) * 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) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    if (k_m <= 3.2d-85) then
        tmp = (2.0d0 / (k_m * (k_m * t))) * ((l / k_m) ** 2.0d0)
    else if (k_m <= 1.5d+124) then
        tmp = ((((l / (k_m * k_m)) * (cos(k_m) / t)) * l) / t_1) * 2.0d0
    else
        tmp = (((cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0d0) / ((t_1 * t) * 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 t_1 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 3.2e-85) {
		tmp = (2.0 / (k_m * (k_m * t))) * Math.pow((l / k_m), 2.0);
	} else if (k_m <= 1.5e+124) {
		tmp = ((((l / (k_m * k_m)) * (Math.cos(k_m) / t)) * l) / t_1) * 2.0;
	} else {
		tmp = (((Math.cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / ((t_1 * t) * 2.0);
	}
	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 <= 3.2e-85:
		tmp = (2.0 / (k_m * (k_m * t))) * math.pow((l / k_m), 2.0)
	elif k_m <= 1.5e+124:
		tmp = ((((l / (k_m * k_m)) * (math.cos(k_m) / t)) * l) / t_1) * 2.0
	else:
		tmp = (((math.cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / ((t_1 * t) * 2.0)
	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 <= 3.2e-85)
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * (Float64(l / k_m) ^ 2.0));
	elseif (k_m <= 1.5e+124)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) * Float64(cos(k_m) / t)) * l) / t_1) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * Float64(l / k_m)) * Float64(l / k_m)) * 4.0) / Float64(Float64(t_1 * t) * 2.0));
	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 <= 3.2e-85)
		tmp = (2.0 / (k_m * (k_m * t))) * ((l / k_m) ^ 2.0);
	elseif (k_m <= 1.5e+124)
		tmp = ((((l / (k_m * k_m)) * (cos(k_m) / t)) * l) / t_1) * 2.0;
	else
		tmp = (((cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / ((t_1 * t) * 2.0);
	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, 3.2e-85], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.5e+124], N[(N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(t$95$1 * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.20000000000000027e-85

   1. Initial program 42.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot {\ell}^{2} - \frac{-1}{3} \cdot {\ell}^{2}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      4. distribute-rgt-out-- N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right), {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, {\ell}^{2}\right)}{{k}^{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, {\ell}^{2}\right)}{{k}^{2}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{{k}^{2}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{{k}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{k \cdot k} \]

      14. lift-*.f64 36.3

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot -0.16666666666666666, k \cdot k, \ell \cdot \ell\right)}{k \cdot k} \]

   10. Applied rewrites 36.3%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot -0.16666666666666666, k \cdot k, \ell \cdot \ell\right)}{\color{blue}{k \cdot k}} \]

   11. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. frac-times N/A

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

       4. pow2 N/A

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

       5. lower-pow.f64 N/A

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

       6. lift-/.f64 78.0

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

   13. Applied rewrites 78.0%

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

   if 3.20000000000000027e-85 < k < 1.5e124

   1. Initial program 39.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 84.9

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

   5. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 99.5%

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

   if 1.5e124 < k

   1. Initial program 41.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 60.6

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 90.6

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

   9. Applied rewrites 90.6%

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

Alternative 2: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 7.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{t}\right) \cdot \ell}{t\_1} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot t} \cdot \left(\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= k_m 3.2e-85)
     (* (/ 2.0 (* k_m (* k_m t))) (pow (/ l k_m) 2.0))
     (if (<= k_m 7.2e+126)
       (* (/ (* (* (/ l (* k_m k_m)) (/ (cos k_m) t)) l) t_1) 2.0)
       (* (/ 2.0 (* t_1 t)) (* (/ (* (cos k_m) l) k_m) (/ l k_m)))))))

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 <= 3.2e-85) {
		tmp = (2.0 / (k_m * (k_m * t))) * pow((l / k_m), 2.0);
	} else if (k_m <= 7.2e+126) {
		tmp = ((((l / (k_m * k_m)) * (cos(k_m) / t)) * l) / t_1) * 2.0;
	} else {
		tmp = (2.0 / (t_1 * t)) * (((cos(k_m) * 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 = sin(k_m) ** 2.0d0
    if (k_m <= 3.2d-85) then
        tmp = (2.0d0 / (k_m * (k_m * t))) * ((l / k_m) ** 2.0d0)
    else if (k_m <= 7.2d+126) then
        tmp = ((((l / (k_m * k_m)) * (cos(k_m) / t)) * l) / t_1) * 2.0d0
    else
        tmp = (2.0d0 / (t_1 * t)) * (((cos(k_m) * 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 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 3.2e-85) {
		tmp = (2.0 / (k_m * (k_m * t))) * Math.pow((l / k_m), 2.0);
	} else if (k_m <= 7.2e+126) {
		tmp = ((((l / (k_m * k_m)) * (Math.cos(k_m) / t)) * l) / t_1) * 2.0;
	} else {
		tmp = (2.0 / (t_1 * t)) * (((Math.cos(k_m) * l) / k_m) * (l / k_m));
	}
	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 <= 3.2e-85:
		tmp = (2.0 / (k_m * (k_m * t))) * math.pow((l / k_m), 2.0)
	elif k_m <= 7.2e+126:
		tmp = ((((l / (k_m * k_m)) * (math.cos(k_m) / t)) * l) / t_1) * 2.0
	else:
		tmp = (2.0 / (t_1 * t)) * (((math.cos(k_m) * l) / k_m) * (l / k_m))
	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 <= 3.2e-85)
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * (Float64(l / k_m) ^ 2.0));
	elseif (k_m <= 7.2e+126)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) * Float64(cos(k_m) / t)) * l) / t_1) * 2.0);
	else
		tmp = Float64(Float64(2.0 / Float64(t_1 * t)) * Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(l / k_m)));
	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 <= 3.2e-85)
		tmp = (2.0 / (k_m * (k_m * t))) * ((l / k_m) ^ 2.0);
	elseif (k_m <= 7.2e+126)
		tmp = ((((l / (k_m * k_m)) * (cos(k_m) / t)) * l) / t_1) * 2.0;
	else
		tmp = (2.0 / (t_1 * t)) * (((cos(k_m) * 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[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 3.2e-85], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7.2e+126], N[(N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(2.0 / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.20000000000000027e-85

   1. Initial program 42.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot {\ell}^{2} - \frac{-1}{3} \cdot {\ell}^{2}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      4. distribute-rgt-out-- N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right), {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, {\ell}^{2}\right)}{{k}^{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, {\ell}^{2}\right)}{{k}^{2}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{{k}^{2}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{{k}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{k \cdot k} \]

      14. lift-*.f64 36.3

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot -0.16666666666666666, k \cdot k, \ell \cdot \ell\right)}{k \cdot k} \]

   10. Applied rewrites 36.3%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot -0.16666666666666666, k \cdot k, \ell \cdot \ell\right)}{\color{blue}{k \cdot k}} \]

   11. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. frac-times N/A

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

       4. pow2 N/A

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

       5. lower-pow.f64 N/A

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

       6. lift-/.f64 78.0

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

   13. Applied rewrites 78.0%

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

   if 3.20000000000000027e-85 < k < 7.2000000000000001e126

   1. Initial program 39.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 99.4%

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

   if 7.2000000000000001e126 < k

   1. Initial program 41.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 58.6

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

   5. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 90.2%

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

Alternative 3: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ t_2 := \cos k\_m \cdot \ell\\ \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{t\_2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{t\_1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot t} \cdot \left(\frac{t\_2}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)) (t_2 (* (cos k_m) l)))
   (if (<= l 2.5e+39)
     (* (* (/ t_2 (* (* k_m t) k_m)) (/ l t_1)) 2.0)
     (* (/ 2.0 (* t_1 t)) (* (/ t_2 k_m) (/ l k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double t_2 = cos(k_m) * l;
	double tmp;
	if (l <= 2.5e+39) {
		tmp = ((t_2 / ((k_m * t) * k_m)) * (l / t_1)) * 2.0;
	} else {
		tmp = (2.0 / (t_1 * t)) * ((t_2 / 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) :: t_2
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    t_2 = cos(k_m) * l
    if (l <= 2.5d+39) then
        tmp = ((t_2 / ((k_m * t) * k_m)) * (l / t_1)) * 2.0d0
    else
        tmp = (2.0d0 / (t_1 * t)) * ((t_2 / 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 = Math.pow(Math.sin(k_m), 2.0);
	double t_2 = Math.cos(k_m) * l;
	double tmp;
	if (l <= 2.5e+39) {
		tmp = ((t_2 / ((k_m * t) * k_m)) * (l / t_1)) * 2.0;
	} else {
		tmp = (2.0 / (t_1 * t)) * ((t_2 / k_m) * (l / k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	t_2 = math.cos(k_m) * l
	tmp = 0
	if l <= 2.5e+39:
		tmp = ((t_2 / ((k_m * t) * k_m)) * (l / t_1)) * 2.0
	else:
		tmp = (2.0 / (t_1 * t)) * ((t_2 / k_m) * (l / k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	t_2 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (l <= 2.5e+39)
		tmp = Float64(Float64(Float64(t_2 / Float64(Float64(k_m * t) * k_m)) * Float64(l / t_1)) * 2.0);
	else
		tmp = Float64(Float64(2.0 / Float64(t_1 * t)) * Float64(Float64(t_2 / k_m) * Float64(l / k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0;
	t_2 = cos(k_m) * l;
	tmp = 0.0;
	if (l <= 2.5e+39)
		tmp = ((t_2 / ((k_m * t) * k_m)) * (l / t_1)) * 2.0;
	else
		tmp = (2.0 / (t_1 * t)) * ((t_2 / 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[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[l, 2.5e+39], N[(N[(N[(t$95$2 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(2.0 / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.50000000000000008e39

   1. Initial program 42.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 75.8

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

   5. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 91.5

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

   9. Applied rewrites 91.5%

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

   if 2.50000000000000008e39 < l

   1. Initial program 39.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 63.5

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

   5. Applied rewrites 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 96.1%

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

Alternative 4: 91.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{t\_1}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{{\sin k\_m}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot 4}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= l 5.6e+60)
     (* (* (/ t_1 (* (* k_m t) k_m)) (/ l (pow (sin k_m) 2.0))) 2.0)
     (/
      (* (* (/ t_1 k_m) (/ l k_m)) 4.0)
      (* (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t) 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 (l <= 5.6e+60) {
		tmp = ((t_1 / ((k_m * t) * k_m)) * (l / pow(sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 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) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * l
    if (l <= 5.6d+60) then
        tmp = ((t_1 / ((k_m * t) * k_m)) * (l / (sin(k_m) ** 2.0d0))) * 2.0d0
    else
        tmp = (((t_1 / k_m) * (l / k_m)) * 4.0d0) / (((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t) * 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 t_1 = Math.cos(k_m) * l;
	double tmp;
	if (l <= 5.6e+60) {
		tmp = ((t_1 / ((k_m * t) * k_m)) * (l / Math.pow(Math.sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t) * 2.0);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if l <= 5.6e+60:
		tmp = ((t_1 / ((k_m * t) * k_m)) * (l / math.pow(math.sin(k_m), 2.0))) * 2.0
	else:
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t) * 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 (l <= 5.6e+60)
		tmp = Float64(Float64(Float64(t_1 / Float64(Float64(k_m * t) * k_m)) * Float64(l / (sin(k_m) ^ 2.0))) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(t_1 / k_m) * Float64(l / k_m)) * 4.0) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t) * 2.0));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (l <= 5.6e+60)
		tmp = ((t_1 / ((k_m * t) * k_m)) * (l / (sin(k_m) ^ 2.0))) * 2.0;
	else
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[l, 5.6e+60], N[(N[(N[(t$95$1 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.6e60

   1. Initial program 42.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 76.2

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

   5. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 91.7

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

   9. Applied rewrites 91.7%

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

   if 5.6e60 < l

   1. Initial program 39.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 96.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 95.8

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

   9. Applied rewrites 95.8%

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

Alternative 5: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 4.3 \cdot 10^{+27}:\\ \;\;\;\;\left(\frac{t\_1}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\ell}{{\sin k\_m}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot 4}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 4.3e+27)
     (* (* (/ t_1 (* (* k_m k_m) t)) (/ l (pow (sin k_m) 2.0))) 2.0)
     (/
      (* (* (/ t_1 k_m) (/ l k_m)) 4.0)
      (* (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t) 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 <= 4.3e+27) {
		tmp = ((t_1 / ((k_m * k_m) * t)) * (l / pow(sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 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) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * l
    if (k_m <= 4.3d+27) then
        tmp = ((t_1 / ((k_m * k_m) * t)) * (l / (sin(k_m) ** 2.0d0))) * 2.0d0
    else
        tmp = (((t_1 / k_m) * (l / k_m)) * 4.0d0) / (((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t) * 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 t_1 = Math.cos(k_m) * l;
	double tmp;
	if (k_m <= 4.3e+27) {
		tmp = ((t_1 / ((k_m * k_m) * t)) * (l / Math.pow(Math.sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t) * 2.0);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if k_m <= 4.3e+27:
		tmp = ((t_1 / ((k_m * k_m) * t)) * (l / math.pow(math.sin(k_m), 2.0))) * 2.0
	else:
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t) * 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 <= 4.3e+27)
		tmp = Float64(Float64(Float64(t_1 / Float64(Float64(k_m * k_m) * t)) * Float64(l / (sin(k_m) ^ 2.0))) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(t_1 / k_m) * Float64(l / k_m)) * 4.0) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t) * 2.0));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (k_m <= 4.3e+27)
		tmp = ((t_1 / ((k_m * k_m) * t)) * (l / (sin(k_m) ^ 2.0))) * 2.0;
	else
		tmp = (((t_1 / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 4.3e+27], N[(N[(N[(t$95$1 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.30000000000000008e27

   1. Initial program 41.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.5%

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

   if 4.30000000000000008e27 < k

   1. Initial program 44.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 90.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 89.9

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

   9. Applied rewrites 89.9%

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

Alternative 6: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{{\sin k\_m}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot 4}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.8e-5)
   (* (* (/ (/ l (* k_m k_m)) t) (/ l (pow (sin k_m) 2.0))) 2.0)
   (/
    (* (* (/ (* (cos k_m) l) k_m) (/ l k_m)) 4.0)
    (* (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.8e-5) {
		tmp = (((l / (k_m * k_m)) / t) * (l / pow(sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = ((((cos(k_m) * l) / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 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 <= 8.8d-5) then
        tmp = (((l / (k_m * k_m)) / t) * (l / (sin(k_m) ** 2.0d0))) * 2.0d0
    else
        tmp = ((((cos(k_m) * l) / k_m) * (l / k_m)) * 4.0d0) / (((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t) * 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 <= 8.8e-5) {
		tmp = (((l / (k_m * k_m)) / t) * (l / Math.pow(Math.sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = ((((Math.cos(k_m) * l) / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t) * 2.0);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.8e-5:
		tmp = (((l / (k_m * k_m)) / t) * (l / math.pow(math.sin(k_m), 2.0))) * 2.0
	else:
		tmp = ((((math.cos(k_m) * l) / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t) * 2.0)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.8e-5)
		tmp = Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(l / (sin(k_m) ^ 2.0))) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(l / k_m)) * 4.0) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t) * 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 <= 8.8e-5)
		tmp = (((l / (k_m * k_m)) / t) * (l / (sin(k_m) ^ 2.0))) * 2.0;
	else
		tmp = ((((cos(k_m) * l) / k_m) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 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, 8.8e-5], N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.7999999999999998e-5

   1. Initial program 42.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   if 8.7999999999999998e-5 < k

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.8

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 89.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 89.3

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

   9. Applied rewrites 89.3%

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

Alternative 7: 93.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{{\sin k\_m}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\right) \cdot 4}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.8e-5)
   (* (* (/ (/ l (* k_m k_m)) t) (/ l (pow (sin k_m) 2.0))) 2.0)
   (/
    (* (* (* (cos k_m) (/ l k_m)) (/ l k_m)) 4.0)
    (* (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.8e-5) {
		tmp = (((l / (k_m * k_m)) / t) * (l / pow(sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 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 <= 8.8d-5) then
        tmp = (((l / (k_m * k_m)) / t) * (l / (sin(k_m) ** 2.0d0))) * 2.0d0
    else
        tmp = (((cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0d0) / (((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t) * 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 <= 8.8e-5) {
		tmp = (((l / (k_m * k_m)) / t) * (l / Math.pow(Math.sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((Math.cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t) * 2.0);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.8e-5:
		tmp = (((l / (k_m * k_m)) / t) * (l / math.pow(math.sin(k_m), 2.0))) * 2.0
	else:
		tmp = (((math.cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t) * 2.0)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.8e-5)
		tmp = Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(l / (sin(k_m) ^ 2.0))) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * Float64(l / k_m)) * Float64(l / k_m)) * 4.0) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t) * 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 <= 8.8e-5)
		tmp = (((l / (k_m * k_m)) / t) * (l / (sin(k_m) ^ 2.0))) * 2.0;
	else
		tmp = (((cos(k_m) * (l / k_m)) * (l / k_m)) * 4.0) / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * 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, 8.8e-5], N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.7999999999999998e-5

   1. Initial program 42.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   if 8.7999999999999998e-5 < k

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.8

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 89.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 89.5

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

   9. Applied rewrites 89.5%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 89.3

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

   11. Applied rewrites 89.3%

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

Alternative 8: 86.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{{\sin k\_m}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos k\_m \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e-5)
   (* (* (/ (/ l (* k_m k_m)) t) (/ l (pow (sin k_m) 2.0))) 2.0)
   (*
    (*
     (/ (* (cos k_m) l) (* (* k_m k_m) t))
     (/ l (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-5) {
		tmp = (((l / (k_m * k_m)) / t) * (l / pow(sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((cos(k_m) * l) / ((k_m * k_m) * t)) * (l / (0.5 - (0.5 * cos((2.0 * 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 <= 7.5d-5) then
        tmp = (((l / (k_m * k_m)) / t) * (l / (sin(k_m) ** 2.0d0))) * 2.0d0
    else
        tmp = (((cos(k_m) * l) / ((k_m * k_m) * t)) * (l / (0.5d0 - (0.5d0 * cos((2.0d0 * 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 <= 7.5e-5) {
		tmp = (((l / (k_m * k_m)) / t) * (l / Math.pow(Math.sin(k_m), 2.0))) * 2.0;
	} else {
		tmp = (((Math.cos(k_m) * l) / ((k_m * k_m) * t)) * (l / (0.5 - (0.5 * Math.cos((2.0 * k_m)))))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.5e-5:
		tmp = (((l / (k_m * k_m)) / t) * (l / math.pow(math.sin(k_m), 2.0))) * 2.0
	else:
		tmp = (((math.cos(k_m) * l) / ((k_m * k_m) * t)) * (l / (0.5 - (0.5 * math.cos((2.0 * k_m)))))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e-5)
		tmp = Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(l / (sin(k_m) ^ 2.0))) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) / Float64(Float64(k_m * k_m) * t)) * Float64(l / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * 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 <= 7.5e-5)
		tmp = (((l / (k_m * k_m)) / t) * (l / (sin(k_m) ^ 2.0))) * 2.0;
	else
		tmp = (((cos(k_m) * l) / ((k_m * k_m) * t)) * (l / (0.5 - (0.5 * cos((2.0 * 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, 7.5e-5], N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * 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] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999934e-5

   1. Initial program 42.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   if 7.49999999999999934e-5 < k

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 77.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 77.2

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

   9. Applied rewrites 77.2%

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

Alternative 9: 77.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6 \cdot 10^{+100}:\\ \;\;\;\;\left(\frac{\cos k\_m \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m \cdot k\_m}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.6e+100)
   (*
    (*
     (/ (* (cos k_m) l) (* (* k_m k_m) t))
     (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) (* k_m k_m)))
    2.0)
   (* (/ (pow (/ l k_m) 2.0) t) -0.3333333333333333)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e+100) {
		tmp = (((cos(k_m) * l) / ((k_m * k_m) * t)) * (fma(0.3333333333333333, ((k_m * k_m) * l), l) / (k_m * k_m))) * 2.0;
	} else {
		tmp = (pow((l / k_m), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.6e+100)
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) / Float64(Float64(k_m * k_m) * t)) * Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / Float64(k_m * k_m))) * 2.0);
	else
		tmp = Float64(Float64((Float64(l / k_m) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.6e+100], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * 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] * 2.0), $MachinePrecision], N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.6000000000000002e100

   1. Initial program 41.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 75.6

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

   5. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \left(\frac{\cos k \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\mathsf{fma}\left(\frac{1}{3}, \left(k \cdot k\right) \cdot \ell, \ell\right)}{k \cdot k}\right) \cdot 2 \]

      8. lift-*.f64 70.1

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

   10. Applied rewrites 70.1%

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

   if 6.6000000000000002e100 < k

   1. Initial program 44.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 26.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   5. Applied rewrites 26.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      9. lift-*.f64 54.7

         \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   8. Applied rewrites 54.7%

      \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      8. frac-times N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 54.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

       6. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

       7. associate-/r* N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       9. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       10. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

       11. frac-times N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lift-/.f64 60.5

           \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333 \]

   12. Applied rewrites 60.5%

       \[\leadsto \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{k\_m}\right)}^{2}\\ \mathbf{if}\;k\_m \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ l k_m) 2.0)))
   (if (<= k_m 3e+98)
     (* (/ 2.0 (* k_m (* k_m t))) t_1)
     (* (/ t_1 t) -0.3333333333333333))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((l / k_m), 2.0);
	double tmp;
	if (k_m <= 3e+98) {
		tmp = (2.0 / (k_m * (k_m * t))) * t_1;
	} else {
		tmp = (t_1 / t) * -0.3333333333333333;
	}
	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 = (l / k_m) ** 2.0d0
    if (k_m <= 3d+98) then
        tmp = (2.0d0 / (k_m * (k_m * t))) * t_1
    else
        tmp = (t_1 / t) * (-0.3333333333333333d0)
    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((l / k_m), 2.0);
	double tmp;
	if (k_m <= 3e+98) {
		tmp = (2.0 / (k_m * (k_m * t))) * t_1;
	} else {
		tmp = (t_1 / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow((l / k_m), 2.0)
	tmp = 0
	if k_m <= 3e+98:
		tmp = (2.0 / (k_m * (k_m * t))) * t_1
	else:
		tmp = (t_1 / t) * -0.3333333333333333
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 3e+98)
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * t_1);
	else
		tmp = Float64(Float64(t_1 / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (l / k_m) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 3e+98)
		tmp = (2.0 / (k_m * (k_m * t))) * t_1;
	else
		tmp = (t_1 / t) * -0.3333333333333333;
	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[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 3e+98], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$1 / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0000000000000001e98

   1. Initial program 41.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 75.5

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

   5. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\frac{-1}{2} \cdot {\ell}^{2} - \frac{-1}{3} \cdot {\ell}^{2}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      4. distribute-rgt-out-- N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right), {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left({\ell}^{2} \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, {k}^{2}, {\ell}^{2}\right)}{{k}^{2}} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, {\ell}^{2}\right)}{{k}^{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, {\ell}^{2}\right)}{{k}^{2}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{{k}^{2}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{{k}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{6}, k \cdot k, \ell \cdot \ell\right)}{k \cdot k} \]

      14. lift-*.f64 38.3

          \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot -0.16666666666666666, k \cdot k, \ell \cdot \ell\right)}{k \cdot k} \]

   10. Applied rewrites 38.3%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot -0.16666666666666666, k \cdot k, \ell \cdot \ell\right)}{\color{blue}{k \cdot k}} \]

   11. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. frac-times N/A

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

       4. pow2 N/A

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

       5. lower-pow.f64 N/A

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

       6. lift-/.f64 76.9

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

   13. Applied rewrites 76.9%

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

   if 3.0000000000000001e98 < k

   1. Initial program 43.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 26.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      9. lift-*.f64 53.9

         \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   8. Applied rewrites 53.9%

      \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      8. frac-times N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 54.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 54.0%

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

       6. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

       7. associate-/r* N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       9. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       10. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

       11. frac-times N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lift-/.f64 59.6

           \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333 \]

   12. Applied rewrites 59.6%

       \[\leadsto \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e+98)
   (* (/ 2.0 (* (* k_m k_m) t)) (* (/ l k_m) (/ l k_m)))
   (* (/ (pow (/ l k_m) 2.0) t) -0.3333333333333333)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e+98) {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (pow((l / k_m), 2.0) / t) * -0.3333333333333333;
	}
	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 <= 3d+98) then
        tmp = (2.0d0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
    else
        tmp = (((l / k_m) ** 2.0d0) / t) * (-0.3333333333333333d0)
    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 <= 3e+98) {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	} else {
		tmp = (Math.pow((l / k_m), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3e+98:
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
	else:
		tmp = (math.pow((l / k_m), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e+98)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64((Float64(l / k_m) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3e+98)
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	else
		tmp = (((l / k_m) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e+98], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0000000000000001e98

   1. Initial program 41.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 75.2

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

   8. Applied rewrites 75.2%

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

   if 3.0000000000000001e98 < k

   1. Initial program 43.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 26.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      9. lift-*.f64 53.9

         \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   8. Applied rewrites 53.9%

      \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      8. frac-times N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 54.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 54.0%

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

       6. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

       7. associate-/r* N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       9. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

       10. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

       11. frac-times N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lift-/.f64 59.6

           \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333 \]

   12. Applied rewrites 59.6%

       \[\leadsto \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* (/ 2.0 (* (* k_m k_m) (* k_m k_m))) (* l (/ l t)))
   (* (/ 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) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * (l * (l / t));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = (2.0d0 / ((k_m * k_m) * (k_m * k_m))) * (l * (l / t))
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * (l * (l / t));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * (l * (l / t))
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m))
	return tmp

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 26.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-*.f64 54.5

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

      5. lower-/.f64 74.0

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   7. Applied rewrites 74.0%

      \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{{k}^{\left(2 + 2\right)}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      3. pow-prod-up N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      8. lift-*.f64 73.9

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

   9. Applied rewrites 73.9%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

   if 0.0 < (*.f64 l l)

   1. Initial program 46.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 79.0

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lower-*.f64 81.9

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}{{\sin k}^{2}} \]

   7. Applied rewrites 81.9%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{{\ell}^{2}}{{k}^{\color{blue}{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

      5. lift-*.f64 65.8

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   10. Applied rewrites 65.8%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 66.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\ell}{k\_m \cdot k\_m}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e+98)
   (* (/ 2.0 (* k_m (* k_m t))) (/ (* l l) (* k_m k_m)))
   (* (* l (/ (/ l (* k_m k_m)) t)) -0.3333333333333333)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e+98) {
		tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
	} else {
		tmp = (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333;
	}
	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 <= 3d+98) then
        tmp = (2.0d0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m))
    else
        tmp = (l * ((l / (k_m * k_m)) / t)) * (-0.3333333333333333d0)
    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 <= 3e+98) {
		tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
	} else {
		tmp = (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3e+98:
		tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m))
	else:
		tmp = (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e+98)
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(l * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(l * Float64(Float64(l / Float64(k_m * k_m)) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3e+98)
		tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
	else
		tmp = (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e+98], N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0000000000000001e98

   1. Initial program 41.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 75.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lower-*.f64 76.5

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}{{\sin k}^{2}} \]

   7. Applied rewrites 76.5%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{{\ell}^{2}}{{k}^{\color{blue}{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

      5. lift-*.f64 65.9

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   10. Applied rewrites 65.9%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

   if 3.0000000000000001e98 < k

   1. Initial program 43.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 26.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      9. lift-*.f64 53.9

         \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   8. Applied rewrites 53.9%

      \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

      8. frac-times N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 54.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 54.0%

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. associate-/l* N/A

          \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

       7. pow2 N/A

          \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{-1}{3} \]

       8. associate-/r* N/A

          \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{2}}}{t}\right) \cdot \frac{-1}{3} \]

       9. lower-/.f64 N/A

          \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{2}}}{t}\right) \cdot \frac{-1}{3} \]

       10. lower-/.f64 N/A

           \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{2}}}{t}\right) \cdot \frac{-1}{3} \]

       11. pow2 N/A

           \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right) \cdot \frac{-1}{3} \]

       12. lift-*.f64 59.2

           \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right) \cdot -0.3333333333333333 \]

   12. Applied rewrites 59.2%

       \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right) \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ 2.0 (* (* k_m k_m) t)) (* (/ l k_m) (/ l k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(l / k_m) * Float64(l / k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 73.2

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 73.2%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
7. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   6. lower-/.f64 71.6

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

8. Applied rewrites 71.6%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]
9. Add Preprocessing

Alternative 15: 30.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \frac{\frac{\ell}{k\_m \cdot k\_m}}{t}\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* l (/ (/ l (* k_m k_m)) t)) -0.3333333333333333))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * ((l / (k_m * k_m)) / t)) * (-0.3333333333333333d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * Float64(Float64(l / Float64(k_m * k_m)) / t)) * -0.3333333333333333)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * ((l / (k_m * k_m)) / t)) * -0.3333333333333333;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   4. div-add-rev N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   8. pow-prod-down N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   14. lower-pow.f64 49.6

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

5. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. times-frac N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   9. lift-*.f64 24.1

      \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 24.1%

   \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

   8. frac-times N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   16. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   17. lift-*.f64 24.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

10. Applied rewrites 24.1%

    \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    3. associate-/l* N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    6. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    7. pow2 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{-1}{3} \]

    8. associate-/r* N/A

       \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{2}}}{t}\right) \cdot \frac{-1}{3} \]

    9. lower-/.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{2}}}{t}\right) \cdot \frac{-1}{3} \]

    10. lower-/.f64 N/A

        \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{2}}}{t}\right) \cdot \frac{-1}{3} \]

    11. pow2 N/A

        \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right) \cdot \frac{-1}{3} \]

    12. lift-*.f64 25.4

        \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right) \cdot -0.3333333333333333 \]

12. Applied rewrites 25.4%

    \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right) \cdot -0.3333333333333333 \]
13. Add Preprocessing

Alternative 16: 30.2% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot \ell}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot -0.3333333333333333 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* l l) (* (* k_m t) k_m)) -0.3333333333333333))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * t) * k_m)) * -0.3333333333333333;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) / ((k_m * t) * k_m)) * (-0.3333333333333333d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * t) * k_m)) * -0.3333333333333333;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) / ((k_m * t) * k_m)) * -0.3333333333333333

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) / Float64(Float64(k_m * t) * k_m)) * -0.3333333333333333)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) / ((k_m * t) * k_m)) * -0.3333333333333333;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   4. div-add-rev N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   8. pow-prod-down N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   14. lower-pow.f64 49.6

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

5. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. times-frac N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   9. lift-*.f64 24.1

      \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 24.1%

   \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

   8. frac-times N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   16. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   17. lift-*.f64 24.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

10. Applied rewrites 24.1%

    \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    3. associate-*r* N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{-1}{3} \]

    4. *-commutative N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot k} \cdot \frac{-1}{3} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot k} \cdot \frac{-1}{3} \]

    6. lift-*.f64 24.7

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot k} \cdot -0.3333333333333333 \]

12. Applied rewrites 24.7%

    \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot k} \cdot -0.3333333333333333 \]
13. Add Preprocessing

Alternative 17: 29.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot -0.3333333333333333 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* l l) (* (* k_m k_m) t)) -0.3333333333333333))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) / ((k_m * k_m) * t)) * (-0.3333333333333333d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * t)) * -0.3333333333333333)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   4. div-add-rev N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   8. pow-prod-down N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   14. lower-pow.f64 49.6

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

5. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. times-frac N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   9. lift-*.f64 24.1

      \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 24.1%

   \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

   8. frac-times N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   16. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   17. lift-*.f64 24.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

10. Applied rewrites 24.1%

    \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))