Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 94.5%

Time: 9.1s

Alternatives: 17

Speedup: 11.0×

• 40.0% 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: 35.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \frac{t}{\ell}\right)\right) \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.2e-66)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (/
    2.0
    (* (* (/ k_m l) (* k_m (/ t l))) (/ (pow (sin k_m) 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e-66) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * (pow(sin(k_m), 2.0) / cos(k_m)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.2d-66) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else
        tmp = 2.0d0 / (((k_m / l) * (k_m * (t / l))) * ((sin(k_m) ** 2.0d0) / cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e-66) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.2e-66:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	else:
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e-66)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(k_m * Float64(t / l))) * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e-66)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	else
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * ((sin(k_m) ^ 2.0) / cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.2e-66], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2000000000000001e-66

   1. Initial program 39.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 80.8

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

   9. Applied rewrites 80.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-/r* N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if 4.2000000000000001e-66 < k

   1. Initial program 29.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 98.8

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

   9. Applied rewrites 98.8%

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

Alternative 2: 93.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \frac{t}{\ell}\right)\right) \cdot \frac{0.5 - \cos \left(k\_m \cdot 2\right) \cdot 0.5}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.32e-5)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (/
    2.0
    (*
     (* (/ k_m l) (* k_m (/ t l)))
     (/ (- 0.5 (* (cos (* k_m 2.0)) 0.5)) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.32e-5) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * ((0.5 - (cos((k_m * 2.0)) * 0.5)) / cos(k_m)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.32d-5) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else
        tmp = 2.0d0 / (((k_m / l) * (k_m * (t / l))) * ((0.5d0 - (cos((k_m * 2.0d0)) * 0.5d0)) / cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.32e-5) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * ((0.5 - (Math.cos((k_m * 2.0)) * 0.5)) / Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.32e-5:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	else:
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * ((0.5 - (math.cos((k_m * 2.0)) * 0.5)) / math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.32e-5)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(k_m * Float64(t / l))) * Float64(Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) * 0.5)) / cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.32e-5)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	else
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * ((0.5 - (cos((k_m * 2.0)) * 0.5)) / cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.32e-5], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.32000000000000007e-5

   1. Initial program 38.8%

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

   3. Taylor expanded in 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 70.4

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

   5. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 72.1

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

   7. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 81.1

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

   9. Applied rewrites 81.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-/r* N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if 1.32000000000000007e-5 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 93.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 98.7

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

   9. Applied rewrites 98.7%

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

       1. lift-sin.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. unpow2 N/A

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

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

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

       5. lower--.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-cos.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

Alternative 3: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.38 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot t\right)}{\cos k\_m} \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.38e-5)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (/
    2.0
    (*
     (/ (* k_m (* k_m t)) (cos k_m))
     (/ (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.38e-5) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m * (k_m * t)) / cos(k_m)) * ((0.5 - (0.5 * cos((2.0 * k_m)))) / (l * l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.38d-5) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else
        tmp = 2.0d0 / (((k_m * (k_m * t)) / cos(k_m)) * ((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) / (l * l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.38e-5) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m * (k_m * t)) / Math.cos(k_m)) * ((0.5 - (0.5 * Math.cos((2.0 * k_m)))) / (l * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.38e-5:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	else:
		tmp = 2.0 / (((k_m * (k_m * t)) / math.cos(k_m)) * ((0.5 - (0.5 * math.cos((2.0 * k_m)))) / (l * l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.38e-5)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m * t)) / cos(k_m)) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) / Float64(l * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.38e-5)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	else
		tmp = 2.0 / (((k_m * (k_m * t)) / cos(k_m)) * ((0.5 - (0.5 * cos((2.0 * k_m)))) / (l * l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.38e-5], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.38e-5

   1. Initial program 38.8%

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

   3. Taylor expanded in 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 70.4

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

   5. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 72.1

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

   7. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 81.1

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

   9. Applied rewrites 81.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-/r* N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if 1.38e-5 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.3

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

   9. Applied rewrites 75.3%

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

Alternative 4: 82.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.38 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{t}{\cos k\_m}\right) \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.38e-5)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (/
    2.0
    (*
     (* (* k_m k_m) (/ t (cos k_m)))
     (/ (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.38e-5) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * (t / cos(k_m))) * ((0.5 - (0.5 * cos((2.0 * k_m)))) / (l * l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.38d-5) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else
        tmp = 2.0d0 / (((k_m * k_m) * (t / cos(k_m))) * ((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) / (l * l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.38e-5) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * (t / Math.cos(k_m))) * ((0.5 - (0.5 * Math.cos((2.0 * k_m)))) / (l * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.38e-5:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	else:
		tmp = 2.0 / (((k_m * k_m) * (t / math.cos(k_m))) * ((0.5 - (0.5 * math.cos((2.0 * k_m)))) / (l * l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.38e-5)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(t / cos(k_m))) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) / Float64(l * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.38e-5)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	else
		tmp = 2.0 / (((k_m * k_m) * (t / cos(k_m))) * ((0.5 - (0.5 * cos((2.0 * k_m)))) / (l * l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.38e-5], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.38e-5

   1. Initial program 38.8%

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

   3. Taylor expanded in 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 70.4

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

   5. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 72.1

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

   7. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 81.1

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

   9. Applied rewrites 81.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-/r* N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if 1.38e-5 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-cos.f64 70.5

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

   9. Applied rewrites 70.5%

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

Alternative 5: 82.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.38 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\cos k\_m} \cdot \frac{0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.38e-5)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (/
    2.0
    (*
     (/ (* (* k_m k_m) t) (cos k_m))
     (/ (- 0.5 (* 0.5 (cos (+ k_m k_m)))) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.38e-5) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / cos(k_m)) * ((0.5 - (0.5 * cos((k_m + k_m)))) / (l * l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.38d-5) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else
        tmp = 2.0d0 / ((((k_m * k_m) * t) / cos(k_m)) * ((0.5d0 - (0.5d0 * cos((k_m + k_m)))) / (l * l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.38e-5) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / Math.cos(k_m)) * ((0.5 - (0.5 * Math.cos((k_m + k_m)))) / (l * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.38e-5:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	else:
		tmp = 2.0 / ((((k_m * k_m) * t) / math.cos(k_m)) * ((0.5 - (0.5 * math.cos((k_m + k_m)))) / (l * l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.38e-5)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) / cos(k_m)) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) / Float64(l * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.38e-5)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	else
		tmp = 2.0 / ((((k_m * k_m) * t) / cos(k_m)) * ((0.5 - (0.5 * cos((k_m + k_m)))) / (l * l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.38e-5], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.38e-5

   1. Initial program 38.8%

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

   3. Taylor expanded in 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 70.4

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

   5. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 72.1

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

   7. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 81.1

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

   9. Applied rewrites 81.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-/r* N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if 1.38e-5 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 70.4

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

   9. Applied rewrites 70.4%

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

Alternative 6: 76.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \frac{t}{\ell}\right)\right) \cdot \frac{{\sin k\_m}^{2}}{1}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.2e-66)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (/ 2.0 (* (* (/ k_m l) (* k_m (/ t l))) (/ (pow (sin k_m) 2.0) 1.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e-66) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * (pow(sin(k_m), 2.0) / 1.0));
	}
	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 <= 4.2d-66) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else
        tmp = 2.0d0 / (((k_m / l) * (k_m * (t / l))) * ((sin(k_m) ** 2.0d0) / 1.0d0))
    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 <= 4.2e-66) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * (Math.pow(Math.sin(k_m), 2.0) / 1.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.2e-66:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	else:
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * (math.pow(math.sin(k_m), 2.0) / 1.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e-66)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(k_m * Float64(t / l))) * Float64((sin(k_m) ^ 2.0) / 1.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e-66)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	else
		tmp = 2.0 / (((k_m / l) * (k_m * (t / l))) * ((sin(k_m) ^ 2.0) / 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.2e-66], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2000000000000001e-66

   1. Initial program 39.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 80.8

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

   9. Applied rewrites 80.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-/r* N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if 4.2000000000000001e-66 < k

   1. Initial program 29.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 98.8

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

   9. Applied rewrites 98.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 58.5%

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

Alternative 7: 75.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{2}{k\_m} \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\cos k\_m} \cdot \frac{k\_m \cdot k\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 5e-276)
   (/ (* (/ 2.0 k_m) (pow (/ l k_m) 2.0)) (* k_m t))
   (/ 2.0 (* (/ (* (* k_m k_m) t) (cos k_m)) (/ (* k_m k_m) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-276) {
		tmp = ((2.0 / k_m) * pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / cos(k_m)) * ((k_m * k_m) / (l * l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d-276) then
        tmp = ((2.0d0 / k_m) * ((l / k_m) ** 2.0d0)) / (k_m * t)
    else
        tmp = 2.0d0 / ((((k_m * k_m) * t) / cos(k_m)) * ((k_m * k_m) / (l * l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-276) {
		tmp = ((2.0 / k_m) * Math.pow((l / k_m), 2.0)) / (k_m * t);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / Math.cos(k_m)) * ((k_m * k_m) / (l * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 5e-276:
		tmp = ((2.0 / k_m) * math.pow((l / k_m), 2.0)) / (k_m * t)
	else:
		tmp = 2.0 / ((((k_m * k_m) * t) / math.cos(k_m)) * ((k_m * k_m) / (l * l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e-276)
		tmp = Float64(Float64(Float64(2.0 / k_m) * (Float64(l / k_m) ^ 2.0)) / Float64(k_m * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) / cos(k_m)) * Float64(Float64(k_m * k_m) / Float64(l * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e-276)
		tmp = ((2.0 / k_m) * ((l / k_m) ^ 2.0)) / (k_m * t);
	else
		tmp = 2.0 / ((((k_m * k_m) * t) / cos(k_m)) * ((k_m * k_m) / (l * l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-276], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999967e-276

   1. Initial program 21.6%

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

   3. Taylor expanded in 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 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 88.2

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

   9. Applied rewrites 88.2%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-/r* N/A

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

       7. frac-times N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 97.3

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

   11. Applied rewrites 97.3%

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

   if 4.99999999999999967e-276 < (*.f64 l l)

   1. Initial program 41.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 72.3

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

   8. Applied rewrites 72.3%

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

Alternative 8: 73.8% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in 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 65.2

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

5. Applied rewrites 65.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 66.5

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

7. Applied rewrites 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. pow2 N/A

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

   16. times-frac N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 72.8

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

9. Applied rewrites 72.8%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. associate-/l* N/A

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

    6. associate-/r* N/A

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

    7. frac-times N/A

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

    8. lower-/.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lower-pow.f64 N/A

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

    14. lower-*.f64 75.3

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

11. Applied rewrites 75.3%

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

Alternative 9: 69.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{\left(\frac{\ell}{k\_m} \cdot \ell\right) \cdot 2}{k\_m \cdot \left(k\_m \cdot k\_m\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{k\_m \cdot k\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 0.0)
   (/ (/ (* (* (/ l k_m) l) 2.0) (* k_m (* k_m k_m))) t)
   (/ 2.0 (* (* (* k_m k_m) t) (/ (* k_m k_m) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = ((((l / k_m) * l) * 2.0) / (k_m * (k_m * k_m))) / t;
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = ((((l / k_m) * l) * 2.0d0) / (k_m * (k_m * k_m))) / t
    else
        tmp = 2.0d0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = ((((l / k_m) * l) * 2.0) / (k_m * (k_m * k_m))) / t;
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = ((((l / k_m) * l) * 2.0) / (k_m * (k_m * k_m))) / t
	else:
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)))
	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(Float64(Float64(Float64(l / k_m) * l) * 2.0) / Float64(k_m * Float64(k_m * k_m))) / t);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m * k_m) / Float64(l * l))));
	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 = ((((l / k_m) * l) * 2.0) / (k_m * (k_m * k_m))) / t;
	else
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)));
	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[(N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 23.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 67.5

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

   7. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 88.0

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

   9. Applied rewrites 88.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. associate-*r/ N/A

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

       8. pow2 N/A

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

       9. frac-times N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 79.6

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

   11. Applied rewrites 79.6%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 40.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 71.1

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 70.0

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

   11. Applied rewrites 70.0%

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

Alternative 10: 72.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m \cdot k\_m}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 7e-39)
   (/ 2.0 (* (* (/ (* k_m k_m) l) (/ t l)) (* k_m k_m)))
   (/ 2.0 (* (* (* k_m k_m) t) (* (/ k_m l) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 7e-39) {
		tmp = 2.0 / ((((k_m * k_m) / l) * (t / l)) * (k_m * k_m));
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 7d-39) then
        tmp = 2.0d0 / ((((k_m * k_m) / l) * (t / l)) * (k_m * k_m))
    else
        tmp = 2.0d0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 7e-39) {
		tmp = 2.0 / ((((k_m * k_m) / l) * (t / l)) * (k_m * k_m));
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 7e-39:
		tmp = 2.0 / ((((k_m * k_m) / l) * (t / l)) * (k_m * k_m))
	else:
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 7e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) / l) * Float64(t / l)) * Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m / l) * Float64(k_m / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 7e-39)
		tmp = 2.0 / ((((k_m * k_m) / l) * (t / l)) * (k_m * k_m));
	else
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 7e-39], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.99999999999999999e-39

   1. Initial program 36.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 76.8

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 76.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      9. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   7. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{\sin k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{k \cdot k}{\cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{\color{blue}{k \cdot k}}{\cos k}} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{k \cdot k}{\cos k}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k \cdot \color{blue}{k}}{\cos k}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k \cdot \color{blue}{k}}{\cos k}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}} \]

      7. lower-/.f64 74.2

         \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}} \]

   10. Applied rewrites 74.2%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{k \cdot k}}{\cos k}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       2. lift-*.f64 71.8

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   13. Applied rewrites 71.8%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   if 6.99999999999999999e-39 < t

   1. Initial program 37.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 82.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 82.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      3. lift-*.f64 77.9

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   8. Applied rewrites 77.9%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{k}^{2}}{\color{blue}{{\ell}^{2}}}} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{{\ell}^{2}}} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\color{blue}{\ell}}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\color{blue}{\ell}}\right)} \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

       6. lift-/.f64 76.7

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   11. Applied rewrites 76.7%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (* (* (/ k_m l) (/ (* k_m t) l)) (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m));
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l)) * Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 78.5

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 78.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   8. pow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

   9. frac-times N/A

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   14. times-frac N/A

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

7. Applied rewrites 94.0%

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 74.0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

10. Applied rewrites 74.0%

    \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]
11. Add Preprocessing

Alternative 12: 71.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (* (* (* k_m k_m) t) (* (/ k_m l) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
}

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) * ((k_m / l) * (k_m / l)))
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) * ((k_m / l) * (k_m / l)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m / l) * Float64(k_m / l))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 78.5

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 78.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]
7. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   3. lift-*.f64 70.4

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

8. Applied rewrites 70.4%

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{k}^{2}}{\color{blue}{{\ell}^{2}}}} \]
10. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{{\ell}^{2}}} \]

    2. pow2 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

    3. times-frac N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\color{blue}{\ell}}\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\color{blue}{\ell}}\right)} \]

    5. lift-/.f64 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

    6. lift-/.f64 73.0

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

11. Applied rewrites 73.0%

    \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}\right)} \]
12. Add Preprocessing

Alternative 13: 64.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{k\_m \cdot k\_m}{\ell \cdot \ell}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (* (* (* k_m k_m) t) (/ (* k_m k_m) (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)));
}

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) * ((k_m * k_m) / (l * l)))
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) * ((k_m * k_m) / (l * l)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m * k_m) / Float64(l * l))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / (((k_m * k_m) * t) * ((k_m * k_m) / (l * l)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 78.5

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 78.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]
7. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   3. lift-*.f64 70.4

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

8. Applied rewrites 70.4%

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]
10. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

    2. lift-*.f64 69.5

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

11. Applied rewrites 69.5%

    \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\color{blue}{\ell} \cdot \ell}} \]
12. Add Preprocessing

Alternative 14: 64.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{2}{k\_m} \cdot \left(\ell \cdot \ell\right)}{k\_m \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (/ 2.0 k_m) (* l l)) (* k_m (* (* k_m k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((2.0 / k_m) * (l * l)) / (k_m * ((k_m * k_m) * t));
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((2.0d0 / k_m) * (l * l)) / (k_m * ((k_m * k_m) * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((2.0 / k_m) * (l * l)) / (k_m * ((k_m * k_m) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((2.0 / k_m) * (l * l)) / (k_m * ((k_m * k_m) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(2.0 / k_m) * Float64(l * l)) / Float64(k_m * Float64(Float64(k_m * k_m) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((2.0 / k_m) * (l * l)) / (k_m * ((k_m * k_m) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 65.2

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

5. Applied rewrites 65.2%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   3. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   6. pow2 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   13. lift-*.f64 66.5

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

7. Applied rewrites 66.5%

   \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{\left(2 + 2\right)}}}{t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot {k}^{2}}}{t} \]

   9. times-frac N/A

      \[\leadsto \frac{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   14. pow2 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2}}}{t} \]

   15. pow2 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{k \cdot k}}{t} \]

   16. times-frac N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

   18. lower-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

   19. lower-/.f64 72.8

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

9. Applied rewrites 72.8%

   \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{\color{blue}{t}} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

    5. associate-/l* N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}} \]

    6. associate-/r* N/A

       \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \]

    8. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \]

    9. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \]

    10. frac-times N/A

        \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \]

    11. pow2 N/A

        \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{\frac{{\ell}^{2}}{k \cdot k}}{t} \]

    12. pow2 N/A

        \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \]

    13. associate-/r* N/A

        \[\leadsto \frac{\frac{2}{k}}{k} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    14. frac-times N/A

        \[\leadsto \frac{\frac{2}{k} \cdot {\ell}^{2}}{\color{blue}{k \cdot \left({k}^{2} \cdot t\right)}} \]

    15. lower-/.f64 N/A

        \[\leadsto \frac{\frac{2}{k} \cdot {\ell}^{2}}{\color{blue}{k \cdot \left({k}^{2} \cdot t\right)}} \]

    16. lower-*.f64 N/A

        \[\leadsto \frac{\frac{2}{k} \cdot {\ell}^{2}}{\color{blue}{k} \cdot \left({k}^{2} \cdot t\right)} \]

    17. lower-/.f64 N/A

        \[\leadsto \frac{\frac{2}{k} \cdot {\ell}^{2}}{k \cdot \left({k}^{2} \cdot t\right)} \]

    18. pow2 N/A

        \[\leadsto \frac{\frac{2}{k} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left({k}^{2} \cdot t\right)} \]

    19. lift-*.f64 N/A

        \[\leadsto \frac{\frac{2}{k} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left({k}^{2} \cdot t\right)} \]

    20. lower-*.f64 N/A

        \[\leadsto \frac{\frac{2}{k} \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

11. Applied rewrites 69.0%

    \[\leadsto \frac{\frac{2}{k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Alternative 15: 63.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (* l l) 2.0) (* (* k_m k_m) (* (* k_m k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * 2.0) / ((k_m * k_m) * ((k_m * k_m) * t));
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) * 2.0d0) / ((k_m * k_m) * ((k_m * k_m) * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) * 2.0) / ((k_m * k_m) * ((k_m * k_m) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) * 2.0) / ((k_m * k_m) * ((k_m * k_m) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(k_m * k_m) * Float64(Float64(k_m * k_m) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) * 2.0) / ((k_m * k_m) * ((k_m * k_m) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 65.2

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

5. Applied rewrites 65.2%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   3. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   6. pow2 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   13. lift-*.f64 66.5

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

7. Applied rewrites 66.5%

   \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{\left(2 + 2\right)}}}{t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot {k}^{2}}}{t} \]

   9. times-frac N/A

      \[\leadsto \frac{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t} \]

   14. pow2 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2}}}{t} \]

   15. pow2 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{k \cdot k}}{t} \]

   16. times-frac N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

   18. lower-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

   19. lower-/.f64 72.8

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

9. Applied rewrites 72.8%

   \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{\color{blue}{t}} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t} \]

    5. associate-/l* N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}} \]

    6. pow2 N/A

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}}{t} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \]

    8. lift-/.f64 N/A

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \]

    9. lift-/.f64 N/A

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \]

    10. frac-times N/A

        \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \]

    11. pow2 N/A

        \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{k \cdot k}}{t} \]

    12. pow2 N/A

        \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \]

    13. associate-/r* N/A

        \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    14. frac-times N/A

        \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

    15. lower-/.f64 N/A

        \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

    16. *-commutative N/A

        \[\leadsto \frac{{\ell}^{2} \cdot 2}{\color{blue}{{k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

    17. lower-*.f64 N/A

        \[\leadsto \frac{{\ell}^{2} \cdot 2}{\color{blue}{{k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

    18. pow2 N/A

        \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

    19. lift-*.f64 N/A

        \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

    20. lower-*.f64 N/A

        \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

11. Applied rewrites 67.8%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Alternative 16: 20.4% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* -0.11666666666666667 (/ (* l l) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.11666666666666667 * ((l * l) / t);
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (-0.11666666666666667d0) * ((l * l) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.11666666666666667 * ((l * l) / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.11666666666666667 * ((l * l) / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.11666666666666667 * Float64(Float64(l * l) / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.11666666666666667 * ((l * l) / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 21.8

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 21.8%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Add Preprocessing

Alternative 17: 18.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.11666666666666667 * (l * (l / t));
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.11666666666666667 * (l * (l / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.11666666666666667 * Float64(l * Float64(l / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.11666666666666667 * (l * (l / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 21.8

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 21.8%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 19.6

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

10. Applied rewrites 19.6%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))