Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 89.2%

Time: 8.3s

Alternatives: 17

Speedup: 5.7×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \cos k \cdot l\_m\\ \mathbf{if}\;l\_m \leq 2.25 \cdot 10^{+167}:\\ \;\;\;\;\left(\frac{\frac{1}{t}}{\sin k} \cdot \frac{2}{\sin k}\right) \cdot \frac{t\_1 \cdot \frac{l\_m}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 \cdot \frac{\frac{l\_m}{k}}{k}\right)}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (cos k) l_m)))
   (if (<= l_m 2.25e+167)
     (* (* (/ (/ 1.0 t) (sin k)) (/ 2.0 (sin k))) (/ (* t_1 (/ l_m k)) k))
     (/ (* 2.0 (* t_1 (/ (/ l_m k) k))) (* (- 0.5 (* (cos (+ k k)) 0.5)) t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = cos(k) * l_m;
	double tmp;
	if (l_m <= 2.25e+167) {
		tmp = (((1.0 / t) / sin(k)) * (2.0 / sin(k))) * ((t_1 * (l_m / k)) / k);
	} else {
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (cos((k + k)) * 0.5)) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l_m
    if (l_m <= 2.25d+167) then
        tmp = (((1.0d0 / t) / sin(k)) * (2.0d0 / sin(k))) * ((t_1 * (l_m / k)) / k)
    else
        tmp = (2.0d0 * (t_1 * ((l_m / k) / k))) / ((0.5d0 - (cos((k + k)) * 0.5d0)) * t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.cos(k) * l_m;
	double tmp;
	if (l_m <= 2.25e+167) {
		tmp = (((1.0 / t) / Math.sin(k)) * (2.0 / Math.sin(k))) * ((t_1 * (l_m / k)) / k);
	} else {
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (Math.cos((k + k)) * 0.5)) * t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.cos(k) * l_m
	tmp = 0
	if l_m <= 2.25e+167:
		tmp = (((1.0 / t) / math.sin(k)) * (2.0 / math.sin(k))) * ((t_1 * (l_m / k)) / k)
	else:
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (math.cos((k + k)) * 0.5)) * t)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (l_m <= 2.25e+167)
		tmp = Float64(Float64(Float64(Float64(1.0 / t) / sin(k)) * Float64(2.0 / sin(k))) * Float64(Float64(t_1 * Float64(l_m / k)) / k));
	else
		tmp = Float64(Float64(2.0 * Float64(t_1 * Float64(Float64(l_m / k) / k))) / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = cos(k) * l_m;
	tmp = 0.0;
	if (l_m <= 2.25e+167)
		tmp = (((1.0 / t) / sin(k)) * (2.0 / sin(k))) * ((t_1 * (l_m / k)) / k);
	else
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (cos((k + k)) * 0.5)) * t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[l$95$m, 2.25e+167], N[(N[(N[(N[(1.0 / t), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$1 * N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.25e167

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-neg-in N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. fp-cancel-sub-sign-inv N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. count-2-rev N/A

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

   10. Applied rewrites 88.4%

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

   if 2.25e167 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

   10. Applied rewrites 80.3%

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

Alternative 2: 88.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \cos k \cdot l\_m\\ t_2 := \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\\ \mathbf{if}\;k \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{+220}:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 \cdot \frac{\frac{l\_m}{k}}{k}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 \cdot \frac{l\_m}{k}\right)}{t\_2 \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (cos k) l_m)) (t_2 (* (- 0.5 (* (cos (+ k k)) 0.5)) t)))
   (if (<= k 4.6e-5)
     (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
     (if (<= k 2.05e+220)
       (/ (* 2.0 (* t_1 (/ (/ l_m k) k))) t_2)
       (/ (* 2.0 (* t_1 (/ l_m k))) (* t_2 k))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = cos(k) * l_m;
	double t_2 = (0.5 - (cos((k + k)) * 0.5)) * t;
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else if (k <= 2.05e+220) {
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / t_2;
	} else {
		tmp = (2.0 * (t_1 * (l_m / k))) / (t_2 * k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(k) * l_m
    t_2 = (0.5d0 - (cos((k + k)) * 0.5d0)) * t
    if (k <= 4.6d-5) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else if (k <= 2.05d+220) then
        tmp = (2.0d0 * (t_1 * ((l_m / k) / k))) / t_2
    else
        tmp = (2.0d0 * (t_1 * (l_m / k))) / (t_2 * k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.cos(k) * l_m;
	double t_2 = (0.5 - (Math.cos((k + k)) * 0.5)) * t;
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else if (k <= 2.05e+220) {
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / t_2;
	} else {
		tmp = (2.0 * (t_1 * (l_m / k))) / (t_2 * k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.cos(k) * l_m
	t_2 = (0.5 - (math.cos((k + k)) * 0.5)) * t
	tmp = 0
	if k <= 4.6e-5:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	elif k <= 2.05e+220:
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / t_2
	else:
		tmp = (2.0 * (t_1 * (l_m / k))) / (t_2 * k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(cos(k) * l_m)
	t_2 = Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t)
	tmp = 0.0
	if (k <= 4.6e-5)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	elseif (k <= 2.05e+220)
		tmp = Float64(Float64(2.0 * Float64(t_1 * Float64(Float64(l_m / k) / k))) / t_2);
	else
		tmp = Float64(Float64(2.0 * Float64(t_1 * Float64(l_m / k))) / Float64(t_2 * k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = cos(k) * l_m;
	t_2 = (0.5 - (cos((k + k)) * 0.5)) * t;
	tmp = 0.0;
	if (k <= 4.6e-5)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	elseif (k <= 2.05e+220)
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / t_2;
	else
		tmp = (2.0 * (t_1 * (l_m / k))) / (t_2 * k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k, 4.6e-5], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.05e+220], N[(N[(2.0 * N[(t$95$1 * N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 * N[(t$95$1 * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * k), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.6e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 4.6e-5 < k < 2.0499999999999999e220

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

   10. Applied rewrites 80.3%

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

   if 2.0499999999999999e220 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.2%

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

Alternative 3: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \cos k \cdot l\_m\\ \mathbf{if}\;l\_m \leq 1.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{t\_1 \cdot \frac{l\_m}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 \cdot \frac{\frac{l\_m}{k}}{k}\right)}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (cos k) l_m)))
   (if (<= l_m 1.5e+187)
     (* (/ 2.0 (* (pow (sin k) 2.0) t)) (/ (* t_1 (/ l_m k)) k))
     (/ (* 2.0 (* t_1 (/ (/ l_m k) k))) (* (- 0.5 (* (cos (+ k k)) 0.5)) t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = cos(k) * l_m;
	double tmp;
	if (l_m <= 1.5e+187) {
		tmp = (2.0 / (pow(sin(k), 2.0) * t)) * ((t_1 * (l_m / k)) / k);
	} else {
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (cos((k + k)) * 0.5)) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l_m
    if (l_m <= 1.5d+187) then
        tmp = (2.0d0 / ((sin(k) ** 2.0d0) * t)) * ((t_1 * (l_m / k)) / k)
    else
        tmp = (2.0d0 * (t_1 * ((l_m / k) / k))) / ((0.5d0 - (cos((k + k)) * 0.5d0)) * t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.cos(k) * l_m;
	double tmp;
	if (l_m <= 1.5e+187) {
		tmp = (2.0 / (Math.pow(Math.sin(k), 2.0) * t)) * ((t_1 * (l_m / k)) / k);
	} else {
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (Math.cos((k + k)) * 0.5)) * t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.cos(k) * l_m
	tmp = 0
	if l_m <= 1.5e+187:
		tmp = (2.0 / (math.pow(math.sin(k), 2.0) * t)) * ((t_1 * (l_m / k)) / k)
	else:
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (math.cos((k + k)) * 0.5)) * t)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (l_m <= 1.5e+187)
		tmp = Float64(Float64(2.0 / Float64((sin(k) ^ 2.0) * t)) * Float64(Float64(t_1 * Float64(l_m / k)) / k));
	else
		tmp = Float64(Float64(2.0 * Float64(t_1 * Float64(Float64(l_m / k) / k))) / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = cos(k) * l_m;
	tmp = 0.0;
	if (l_m <= 1.5e+187)
		tmp = (2.0 / ((sin(k) ^ 2.0) * t)) * ((t_1 * (l_m / k)) / k);
	else
		tmp = (2.0 * (t_1 * ((l_m / k) / k))) / ((0.5 - (cos((k + k)) * 0.5)) * t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[l$95$m, 1.5e+187], N[(N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$1 * N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5e187

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

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

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 87.5

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

   10. Applied rewrites 87.5%

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

   if 1.5e187 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

   10. Applied rewrites 80.3%

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

Alternative 4: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \cdot \frac{\left(\cos k \cdot l\_m\right) \cdot \frac{l\_m}{k}}{k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.6e-5)
   (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
   (*
    (/ 2.0 (* (- 0.5 (* (cos (+ k k)) 0.5)) t))
    (/ (* (* (cos k) l_m) (/ l_m k)) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t)) * (((cos(k) * l_m) * (l_m / k)) / k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d-5) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else
        tmp = (2.0d0 / ((0.5d0 - (cos((k + k)) * 0.5d0)) * t)) * (((cos(k) * l_m) * (l_m / k)) / k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / ((0.5 - (Math.cos((k + k)) * 0.5)) * t)) * (((Math.cos(k) * l_m) * (l_m / k)) / k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 4.6e-5:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	else:
		tmp = (2.0 / ((0.5 - (math.cos((k + k)) * 0.5)) * t)) * (((math.cos(k) * l_m) * (l_m / k)) / k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.6e-5)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t)) * Float64(Float64(Float64(cos(k) * l_m) * Float64(l_m / k)) / k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 4.6e-5)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	else
		tmp = (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t)) * (((cos(k) * l_m) * (l_m / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.6e-5], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 4.6e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

Alternative 5: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \cdot \frac{\cos k \cdot \left(l\_m \cdot \frac{l\_m}{k}\right)}{k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.6e-5)
   (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
   (*
    (/ 2.0 (* (- 0.5 (* (cos (+ k k)) 0.5)) t))
    (/ (* (cos k) (* l_m (/ l_m k))) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t)) * ((cos(k) * (l_m * (l_m / k))) / k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d-5) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else
        tmp = (2.0d0 / ((0.5d0 - (cos((k + k)) * 0.5d0)) * t)) * ((cos(k) * (l_m * (l_m / k))) / k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / ((0.5 - (Math.cos((k + k)) * 0.5)) * t)) * ((Math.cos(k) * (l_m * (l_m / k))) / k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 4.6e-5:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	else:
		tmp = (2.0 / ((0.5 - (math.cos((k + k)) * 0.5)) * t)) * ((math.cos(k) * (l_m * (l_m / k))) / k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.6e-5)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t)) * Float64(Float64(cos(k) * Float64(l_m * Float64(l_m / k))) / k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 4.6e-5)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	else
		tmp = (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t)) * ((cos(k) * (l_m * (l_m / k))) / k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.6e-5], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 4.6e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 79.7

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

   10. Applied rewrites 79.7%

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

Alternative 6: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k \cdot l\_m\right) \cdot \frac{l\_m}{k}\right)}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.6e-5)
   (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
   (/
    (* 2.0 (* (* (cos k) l_m) (/ l_m k)))
    (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 * ((cos(k) * l_m) * (l_m / k))) / (((0.5 - (cos((k + k)) * 0.5)) * t) * k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d-5) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else
        tmp = (2.0d0 * ((cos(k) * l_m) * (l_m / k))) / (((0.5d0 - (cos((k + k)) * 0.5d0)) * t) * k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 * ((Math.cos(k) * l_m) * (l_m / k))) / (((0.5 - (Math.cos((k + k)) * 0.5)) * t) * k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 4.6e-5:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	else:
		tmp = (2.0 * ((math.cos(k) * l_m) * (l_m / k))) / (((0.5 - (math.cos((k + k)) * 0.5)) * t) * k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.6e-5)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) * l_m) * Float64(l_m / k))) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t) * k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 4.6e-5)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	else
		tmp = (2.0 * ((cos(k) * l_m) * (l_m / k))) / (((0.5 - (cos((k + k)) * 0.5)) * t) * k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.6e-5], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 4.6e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.2%

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

Alternative 7: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(l\_m \cdot l\_m\right) \cdot 2}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\cos k}{k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.6e-5)
   (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
   (*
    (/ (* (* l_m l_m) 2.0) (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t) k))
    (/ (cos k) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (((l_m * l_m) * 2.0) / (((0.5 - (cos((k + k)) * 0.5)) * t) * k)) * (cos(k) / k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d-5) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else
        tmp = (((l_m * l_m) * 2.0d0) / (((0.5d0 - (cos((k + k)) * 0.5d0)) * t) * k)) * (cos(k) / k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (((l_m * l_m) * 2.0) / (((0.5 - (Math.cos((k + k)) * 0.5)) * t) * k)) * (Math.cos(k) / k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 4.6e-5:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	else:
		tmp = (((l_m * l_m) * 2.0) / (((0.5 - (math.cos((k + k)) * 0.5)) * t) * k)) * (math.cos(k) / k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.6e-5)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) * 2.0) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t) * k)) * Float64(cos(k) / k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 4.6e-5)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	else
		tmp = (((l_m * l_m) * 2.0) / (((0.5 - (cos((k + k)) * 0.5)) * t) * k)) * (cos(k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.6e-5], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 4.6e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

   6. Applied rewrites 71.3%

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

Alternative 8: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\left(\cos k \cdot l\_m\right) \cdot l\_m}{k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.6e-5)
   (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
   (*
    (/ 2.0 (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t) k))
    (/ (* (* (cos k) l_m) l_m) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / (((0.5 - (cos((k + k)) * 0.5)) * t) * k)) * (((cos(k) * l_m) * l_m) / k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d-5) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else
        tmp = (2.0d0 / (((0.5d0 - (cos((k + k)) * 0.5d0)) * t) * k)) * (((cos(k) * l_m) * l_m) / k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / (((0.5 - (Math.cos((k + k)) * 0.5)) * t) * k)) * (((Math.cos(k) * l_m) * l_m) / k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 4.6e-5:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	else:
		tmp = (2.0 / (((0.5 - (math.cos((k + k)) * 0.5)) * t) * k)) * (((math.cos(k) * l_m) * l_m) / k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.6e-5)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t) * k)) * Float64(Float64(Float64(cos(k) * l_m) * l_m) / k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 4.6e-5)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	else
		tmp = (2.0 / (((0.5 - (cos((k + k)) * 0.5)) * t) * k)) * (((cos(k) * l_m) * l_m) / k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.6e-5], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 4.6e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   6. Applied rewrites 71.4%

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

Alternative 9: 78.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot l\_m\right) \cdot l\_m\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.6e-5)
   (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
   (/
    (* (* (* (cos k) l_m) l_m) 2.0)
    (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t) k) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (((cos(k) * l_m) * l_m) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t) * k) * k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d-5) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else
        tmp = (((cos(k) * l_m) * l_m) * 2.0d0) / ((((0.5d0 - (cos((k + k)) * 0.5d0)) * t) * k) * k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.6e-5) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (((Math.cos(k) * l_m) * l_m) * 2.0) / ((((0.5 - (Math.cos((k + k)) * 0.5)) * t) * k) * k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 4.6e-5:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	else:
		tmp = (((math.cos(k) * l_m) * l_m) * 2.0) / ((((0.5 - (math.cos((k + k)) * 0.5)) * t) * k) * k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.6e-5)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * l_m) * l_m) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t) * k) * k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 4.6e-5)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	else
		tmp = (((cos(k) * l_m) * l_m) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t) * k) * k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.6e-5], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 4.6e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   6. Applied rewrites 69.7%

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

Alternative 10: 75.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \cos k \cdot l\_m\\ \mathbf{if}\;l\_m \leq 1.7 \cdot 10^{+169}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{t\_1 \cdot \frac{l\_m}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k} \cdot \frac{t\_1 \cdot l\_m}{k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (cos k) l_m)))
   (if (<= l_m 1.7e+169)
     (* (/ 2.0 (* (* k k) t)) (/ (* t_1 (/ l_m k)) k))
     (* (/ 2.0 (* (* (- 0.5 0.5) t) k)) (/ (* t_1 l_m) k)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = cos(k) * l_m;
	double tmp;
	if (l_m <= 1.7e+169) {
		tmp = (2.0 / ((k * k) * t)) * ((t_1 * (l_m / k)) / k);
	} else {
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * ((t_1 * l_m) / k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l_m
    if (l_m <= 1.7d+169) then
        tmp = (2.0d0 / ((k * k) * t)) * ((t_1 * (l_m / k)) / k)
    else
        tmp = (2.0d0 / (((0.5d0 - 0.5d0) * t) * k)) * ((t_1 * l_m) / k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.cos(k) * l_m;
	double tmp;
	if (l_m <= 1.7e+169) {
		tmp = (2.0 / ((k * k) * t)) * ((t_1 * (l_m / k)) / k);
	} else {
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * ((t_1 * l_m) / k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.cos(k) * l_m
	tmp = 0
	if l_m <= 1.7e+169:
		tmp = (2.0 / ((k * k) * t)) * ((t_1 * (l_m / k)) / k)
	else:
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * ((t_1 * l_m) / k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (l_m <= 1.7e+169)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(t_1 * Float64(l_m / k)) / k));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(0.5 - 0.5) * t) * k)) * Float64(Float64(t_1 * l_m) / k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = cos(k) * l_m;
	tmp = 0.0;
	if (l_m <= 1.7e+169)
		tmp = (2.0 / ((k * k) * t)) * ((t_1 * (l_m / k)) / k);
	else
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * ((t_1 * l_m) / k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[l$95$m, 1.7e+169], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * l$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.70000000000000014e169

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 79.7%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if 1.70000000000000014e169 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   6. Applied rewrites 71.4%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 35.7%

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

Alternative 11: 72.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.65 \cdot 10^{+156}:\\ \;\;\;\;\frac{2 \cdot l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\left(\cos k \cdot l\_m\right) \cdot l\_m}{k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 1.65e+156)
   (* (/ (* 2.0 l_m) (* k k)) (/ l_m (* (* k k) t)))
   (* (/ 2.0 (* (* (- 0.5 0.5) t) k)) (/ (* (* (cos k) l_m) l_m) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 1.65e+156) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * (((cos(k) * l_m) * l_m) / k);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l_m <= 1.65d+156) then
        tmp = ((2.0d0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
    else
        tmp = (2.0d0 / (((0.5d0 - 0.5d0) * t) * k)) * (((cos(k) * l_m) * l_m) / k)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 1.65e+156) {
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	} else {
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * (((Math.cos(k) * l_m) * l_m) / k);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if l_m <= 1.65e+156:
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t))
	else:
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * (((math.cos(k) * l_m) * l_m) / k)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 1.65e+156)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(k * k)) * Float64(l_m / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(0.5 - 0.5) * t) * k)) * Float64(Float64(Float64(cos(k) * l_m) * l_m) / k));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (l_m <= 1.65e+156)
		tmp = ((2.0 * l_m) / (k * k)) * (l_m / ((k * k) * t));
	else
		tmp = (2.0 / (((0.5 - 0.5) * t) * k)) * (((cos(k) * l_m) * l_m) / k);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 1.65e+156], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6499999999999999e156

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

   if 1.6499999999999999e156 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   6. Applied rewrites 71.4%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 35.7%

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

Alternative 12: 72.3% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.8

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

6. Applied rewrites 72.8%

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

Alternative 13: 71.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2 \cdot l\_m}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{l\_m}{k} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ (* 2.0 l_m) (* (* (* k k) t) k)) (/ l_m k)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return ((2.0 * l_m) / (((k * k) * t) * k)) * (l_m / k);
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = ((2.0d0 * l_m) / (((k * k) * t) * k)) * (l_m / k)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return ((2.0 * l_m) / (((k * k) * t) * k)) * (l_m / k);
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return ((2.0 * l_m) / (((k * k) * t) * k)) * (l_m / k)

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(Float64(2.0 * l_m) / Float64(Float64(Float64(k * k) * t) * k)) * Float64(l_m / k))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = ((2.0 * l_m) / (((k * k) * t) * k)) * (l_m / k);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 62.5

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   3. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. lift-*.f64 64.0

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

6. Applied rewrites 64.0%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left({k}^{2} \cdot t\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

   9. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   11. lift-*.f64 64.0

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k} \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{k} \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right) \cdot \color{blue}{k}} \]

   7. times-frac N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \frac{\ell}{k} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \frac{\ell}{k} \]

   12. *-commutative N/A

       \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   15. pow2 N/A

       \[\leadsto \frac{2 \cdot \ell}{\left({k}^{2} \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{\left({k}^{2} \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   17. pow2 N/A

       \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

   20. lift-/.f64 71.8

       \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}} \]

10. Applied rewrites 71.8%

    \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]
11. Add Preprocessing

Alternative 14: 68.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\frac{2 \cdot l\_m}{\left(\left(k \cdot k\right) \cdot k\right) \cdot k} \cdot \frac{l\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 2e+95)
   (* (/ (* 2.0 l_m) (* (* (* k k) k) k)) (/ l_m t))
   (/ (* 2.0 (* l_m l_m)) (* (* k k) (* k (* k t))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 2e+95) {
		tmp = ((2.0 * l_m) / (((k * k) * k) * k)) * (l_m / t);
	} else {
		tmp = (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2d+95) then
        tmp = ((2.0d0 * l_m) / (((k * k) * k) * k)) * (l_m / t)
    else
        tmp = (2.0d0 * (l_m * l_m)) / ((k * k) * (k * (k * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 2e+95) {
		tmp = ((2.0 * l_m) / (((k * k) * k) * k)) * (l_m / t);
	} else {
		tmp = (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if t <= 2e+95:
		tmp = ((2.0 * l_m) / (((k * k) * k) * k)) * (l_m / t)
	else:
		tmp = (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 2e+95)
		tmp = Float64(Float64(Float64(2.0 * l_m) / Float64(Float64(Float64(k * k) * k) * k)) * Float64(l_m / t));
	else
		tmp = Float64(Float64(2.0 * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (t <= 2e+95)
		tmp = ((2.0 * l_m) / (((k * k) * k) * k)) * (l_m / t);
	else
		tmp = (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 2e+95], N[(N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000004e95

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      13. lower-*.f64 62.5

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. pow-prod-up N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{\left(2 + 2\right)} \cdot t} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{4} \cdot t} \]

      12. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{4}} \cdot \color{blue}{\frac{\ell}{t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{4}} \cdot \color{blue}{\frac{\ell}{t}} \]

   6. Applied rewrites 67.9%

      \[\leadsto \frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t}} \]

   if 2.00000000000000004e95 < t

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      13. lower-*.f64 62.5

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      3. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      8. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      12. lift-*.f64 64.0

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   6. Applied rewrites 64.0%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      5. lower-*.f64 64.0

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   8. Applied rewrites 64.0%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 64.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{\left(2 \cdot l\_m\right) \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ (* (* 2.0 l_m) l_m) (* (* k k) (* (* k k) t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return ((2.0 * l_m) * l_m) / ((k * k) * ((k * k) * t));
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = ((2.0d0 * l_m) * l_m) / ((k * k) * ((k * k) * t))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return ((2.0 * l_m) * l_m) / ((k * k) * ((k * k) * t));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return ((2.0 * l_m) * l_m) / ((k * k) * ((k * k) * t))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(Float64(2.0 * l_m) * l_m) / Float64(Float64(k * k) * Float64(Float64(k * k) * t)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = ((2.0 * l_m) * l_m) / ((k * k) * ((k * k) * t));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(N[(2.0 * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 62.5

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   3. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. lift-*.f64 64.0

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

6. Applied rewrites 64.0%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot \color{blue}{k}\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   5. lower-*.f64 64.0

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\color{blue}{k} \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]
9. Add Preprocessing

Alternative 16: 64.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2 \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ (* 2.0 (* l_m l_m)) (* (* k k) (* k (* k t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)));
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 * (l_m * l_m)) / ((k * k) * (k * (k * t)))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(k * Float64(k * t))))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 * (l_m * l_m)) / ((k * k) * (k * (k * t)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 62.5

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   3. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. lift-*.f64 64.0

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

6. Applied rewrites 64.0%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   5. lower-*.f64 64.0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]
9. Add Preprocessing

Alternative 17: 64.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2 \cdot \left(l\_m \cdot l\_m\right)}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ (* 2.0 (* l_m l_m)) (* k (* k (* k (* k t))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 * (l_m * l_m)) / (k * (k * (k * (k * t))));
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 * (l_m * l_m)) / (k * (k * (k * (k * t))))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (2.0 * (l_m * l_m)) / (k * (k * (k * (k * t))));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 * (l_m * l_m)) / (k * (k * (k * (k * t))))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 * Float64(l_m * l_m)) / Float64(k * Float64(k * Float64(k * Float64(k * t)))))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 * (l_m * l_m)) / (k * (k * (k * (k * t))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 62.5

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   3. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. lift-*.f64 64.0

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

6. Applied rewrites 64.0%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left({k}^{2} \cdot t\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

   9. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   11. lift-*.f64 64.0

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)} \]

   5. lower-*.f64 64.0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)\right)} \]

10. Applied rewrites 64.0%

    \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025138 
(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))))