Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 94.1%

Time: 8.7s

Alternatives: 20

Speedup: 4.4×

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \tan k \cdot t\\ \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_1}{\left|\ell\right|} \cdot \frac{\sin k}{\left|\ell\right|}\right)\right)}\\ \mathbf{elif}\;\left|\ell\right| \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\frac{k}{\left|\ell\right| \cdot \left|\ell\right|} \cdot \left(t\_1 \cdot \left(\sin k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{\frac{k}{\left|\ell\right|} \cdot k}{\left|\ell\right|}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) t)))
   (if (<= (fabs l) 2e-101)
     (/ 2.0 (* k (* k (* (/ t_1 (fabs l)) (/ (sin k) (fabs l))))))
     (if (<= (fabs l) 2e+146)
       (/ 2.0 (* (/ k (* (fabs l) (fabs l))) (* t_1 (* (sin k) k))))
       (/
        2.0
        (* (* t (* (tan k) (sin k))) (/ (* (/ k (fabs l)) k) (fabs l))))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * t;
	double tmp;
	if (fabs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / fabs(l)) * (sin(k) / fabs(l)))));
	} else if (fabs(l) <= 2e+146) {
		tmp = 2.0 / ((k / (fabs(l) * fabs(l))) * (t_1 * (sin(k) * k)));
	} else {
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (((k / fabs(l)) * k) / fabs(l)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = tan(k) * t
    if (abs(l) <= 2d-101) then
        tmp = 2.0d0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))))
    else if (abs(l) <= 2d+146) then
        tmp = 2.0d0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)))
    else
        tmp = 2.0d0 / ((t * (tan(k) * sin(k))) * (((k / abs(l)) * k) / abs(l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(k) * t;
	double tmp;
	if (Math.abs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / Math.abs(l)) * (Math.sin(k) / Math.abs(l)))));
	} else if (Math.abs(l) <= 2e+146) {
		tmp = 2.0 / ((k / (Math.abs(l) * Math.abs(l))) * (t_1 * (Math.sin(k) * k)));
	} else {
		tmp = 2.0 / ((t * (Math.tan(k) * Math.sin(k))) * (((k / Math.abs(l)) * k) / Math.abs(l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(k) * t
	tmp = 0
	if math.fabs(l) <= 2e-101:
		tmp = 2.0 / (k * (k * ((t_1 / math.fabs(l)) * (math.sin(k) / math.fabs(l)))))
	elif math.fabs(l) <= 2e+146:
		tmp = 2.0 / ((k / (math.fabs(l) * math.fabs(l))) * (t_1 * (math.sin(k) * k)))
	else:
		tmp = 2.0 / ((t * (math.tan(k) * math.sin(k))) * (((k / math.fabs(l)) * k) / math.fabs(l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * t)
	tmp = 0.0
	if (abs(l) <= 2e-101)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_1 / abs(l)) * Float64(sin(k) / abs(l))))));
	elseif (abs(l) <= 2e+146)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(abs(l) * abs(l))) * Float64(t_1 * Float64(sin(k) * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(tan(k) * sin(k))) * Float64(Float64(Float64(k / abs(l)) * k) / abs(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(k) * t;
	tmp = 0.0;
	if (abs(l) <= 2e-101)
		tmp = 2.0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))));
	elseif (abs(l) <= 2e+146)
		tmp = 2.0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)));
	else
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (((k / abs(l)) * k) / abs(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 2e-101], N[(2.0 / N[(k * N[(k * N[(N[(t$95$1 / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 2e+146], N[(2.0 / N[(N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / N[Abs[l], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.0000000000000001e-101

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if 2.0000000000000001e-101 < l < 1.99999999999999987e146

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.3

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

   8. Applied rewrites 78.3%

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

   if 1.99999999999999987e146 < l

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.1

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

   8. Applied rewrites 89.1%

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

Alternative 2: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot \tan k\right) \cdot \left(\left(\sin k \cdot k\right) \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{\frac{k}{\ell} \cdot k}{\ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.7e+100)
   (/ 2.0 (/ (* (* (/ k l) (tan k)) (* (* (sin k) k) t)) l))
   (/ 2.0 (* (* t (* (tan k) (sin k))) (/ (* (/ k l) k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.7e+100) {
		tmp = 2.0 / ((((k / l) * tan(k)) * ((sin(k) * k) * t)) / l);
	} else {
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (((k / l) * k) / l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= 1.7d+100) then
        tmp = 2.0d0 / ((((k / l) * tan(k)) * ((sin(k) * k) * t)) / l)
    else
        tmp = 2.0d0 / ((t * (tan(k) * sin(k))) * (((k / l) * k) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.7e+100) {
		tmp = 2.0 / ((((k / l) * Math.tan(k)) * ((Math.sin(k) * k) * t)) / l);
	} else {
		tmp = 2.0 / ((t * (Math.tan(k) * Math.sin(k))) * (((k / l) * k) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.7e+100:
		tmp = 2.0 / ((((k / l) * math.tan(k)) * ((math.sin(k) * k) * t)) / l)
	else:
		tmp = 2.0 / ((t * (math.tan(k) * math.sin(k))) * (((k / l) * k) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.7e+100)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * tan(k)) * Float64(Float64(sin(k) * k) * t)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(tan(k) * sin(k))) * Float64(Float64(Float64(k / l) * k) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.7e+100)
		tmp = 2.0 / ((((k / l) * tan(k)) * ((sin(k) * k) * t)) / l);
	else
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (((k / l) * k) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.7e+100], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.69999999999999997e100

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 91.0

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

   8. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.6

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

   10. Applied rewrites 91.6%

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

   if 1.69999999999999997e100 < t

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.1

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

   8. Applied rewrites 89.1%

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

Alternative 3: 93.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \tan k \cdot t\\ \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_1}{\left|\ell\right|} \cdot \frac{\sin k}{\left|\ell\right|}\right)\right)}\\ \mathbf{elif}\;\left|\ell\right| \leq 1.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\frac{k}{\left|\ell\right| \cdot \left|\ell\right|} \cdot \left(t\_1 \cdot \left(\sin k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\left(\left(k \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{t}{\left|\ell\right|}}{\left|\ell\right|}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) t)))
   (if (<= (fabs l) 2e-101)
     (/ 2.0 (* k (* k (* (/ t_1 (fabs l)) (/ (sin k) (fabs l))))))
     (if (<= (fabs l) 1.6e+142)
       (/ 2.0 (* (/ k (* (fabs l) (fabs l))) (* t_1 (* (sin k) k))))
       (/
        2.0
        (* k (/ (* (* (* k (sin k)) (tan k)) (/ t (fabs l))) (fabs l))))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * t;
	double tmp;
	if (fabs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / fabs(l)) * (sin(k) / fabs(l)))));
	} else if (fabs(l) <= 1.6e+142) {
		tmp = 2.0 / ((k / (fabs(l) * fabs(l))) * (t_1 * (sin(k) * k)));
	} else {
		tmp = 2.0 / (k * ((((k * sin(k)) * tan(k)) * (t / fabs(l))) / fabs(l)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = tan(k) * t
    if (abs(l) <= 2d-101) then
        tmp = 2.0d0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))))
    else if (abs(l) <= 1.6d+142) then
        tmp = 2.0d0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)))
    else
        tmp = 2.0d0 / (k * ((((k * sin(k)) * tan(k)) * (t / abs(l))) / abs(l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(k) * t;
	double tmp;
	if (Math.abs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / Math.abs(l)) * (Math.sin(k) / Math.abs(l)))));
	} else if (Math.abs(l) <= 1.6e+142) {
		tmp = 2.0 / ((k / (Math.abs(l) * Math.abs(l))) * (t_1 * (Math.sin(k) * k)));
	} else {
		tmp = 2.0 / (k * ((((k * Math.sin(k)) * Math.tan(k)) * (t / Math.abs(l))) / Math.abs(l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(k) * t
	tmp = 0
	if math.fabs(l) <= 2e-101:
		tmp = 2.0 / (k * (k * ((t_1 / math.fabs(l)) * (math.sin(k) / math.fabs(l)))))
	elif math.fabs(l) <= 1.6e+142:
		tmp = 2.0 / ((k / (math.fabs(l) * math.fabs(l))) * (t_1 * (math.sin(k) * k)))
	else:
		tmp = 2.0 / (k * ((((k * math.sin(k)) * math.tan(k)) * (t / math.fabs(l))) / math.fabs(l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * t)
	tmp = 0.0
	if (abs(l) <= 2e-101)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_1 / abs(l)) * Float64(sin(k) / abs(l))))));
	elseif (abs(l) <= 1.6e+142)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(abs(l) * abs(l))) * Float64(t_1 * Float64(sin(k) * k))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(Float64(k * sin(k)) * tan(k)) * Float64(t / abs(l))) / abs(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(k) * t;
	tmp = 0.0;
	if (abs(l) <= 2e-101)
		tmp = 2.0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))));
	elseif (abs(l) <= 1.6e+142)
		tmp = 2.0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)));
	else
		tmp = 2.0 / (k * ((((k * sin(k)) * tan(k)) * (t / abs(l))) / abs(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 2e-101], N[(2.0 / N[(k * N[(k * N[(N[(t$95$1 / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 1.6e+142], N[(2.0 / N[(N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.0000000000000001e-101

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if 2.0000000000000001e-101 < l < 1.60000000000000003e142

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.3

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

   8. Applied rewrites 78.3%

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

   if 1.60000000000000003e142 < l

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 88.8

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

   8. Applied rewrites 88.8%

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

Alternative 4: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \tan k \cdot t\\ \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_1}{\left|\ell\right|} \cdot \frac{\sin k}{\left|\ell\right|}\right)\right)}\\ \mathbf{elif}\;\left|\ell\right| \leq 1.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\frac{k}{\left|\ell\right| \cdot \left|\ell\right|} \cdot \left(t\_1 \cdot \left(\sin k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\left|\ell\right|}\right) \cdot \frac{k}{\left|\ell\right|}\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) t)))
   (if (<= (fabs l) 2e-101)
     (/ 2.0 (* k (* k (* (/ t_1 (fabs l)) (/ (sin k) (fabs l))))))
     (if (<= (fabs l) 1.6e+142)
       (/ 2.0 (* (/ k (* (fabs l) (fabs l))) (* t_1 (* (sin k) k))))
       (/
        2.0
        (* k (* (* (* (sin k) (tan k)) (/ t (fabs l))) (/ k (fabs l)))))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * t;
	double tmp;
	if (fabs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / fabs(l)) * (sin(k) / fabs(l)))));
	} else if (fabs(l) <= 1.6e+142) {
		tmp = 2.0 / ((k / (fabs(l) * fabs(l))) * (t_1 * (sin(k) * k)));
	} else {
		tmp = 2.0 / (k * (((sin(k) * tan(k)) * (t / fabs(l))) * (k / fabs(l))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = tan(k) * t
    if (abs(l) <= 2d-101) then
        tmp = 2.0d0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))))
    else if (abs(l) <= 1.6d+142) then
        tmp = 2.0d0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)))
    else
        tmp = 2.0d0 / (k * (((sin(k) * tan(k)) * (t / abs(l))) * (k / abs(l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(k) * t;
	double tmp;
	if (Math.abs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / Math.abs(l)) * (Math.sin(k) / Math.abs(l)))));
	} else if (Math.abs(l) <= 1.6e+142) {
		tmp = 2.0 / ((k / (Math.abs(l) * Math.abs(l))) * (t_1 * (Math.sin(k) * k)));
	} else {
		tmp = 2.0 / (k * (((Math.sin(k) * Math.tan(k)) * (t / Math.abs(l))) * (k / Math.abs(l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(k) * t
	tmp = 0
	if math.fabs(l) <= 2e-101:
		tmp = 2.0 / (k * (k * ((t_1 / math.fabs(l)) * (math.sin(k) / math.fabs(l)))))
	elif math.fabs(l) <= 1.6e+142:
		tmp = 2.0 / ((k / (math.fabs(l) * math.fabs(l))) * (t_1 * (math.sin(k) * k)))
	else:
		tmp = 2.0 / (k * (((math.sin(k) * math.tan(k)) * (t / math.fabs(l))) * (k / math.fabs(l))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * t)
	tmp = 0.0
	if (abs(l) <= 2e-101)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_1 / abs(l)) * Float64(sin(k) / abs(l))))));
	elseif (abs(l) <= 1.6e+142)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(abs(l) * abs(l))) * Float64(t_1 * Float64(sin(k) * k))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(sin(k) * tan(k)) * Float64(t / abs(l))) * Float64(k / abs(l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(k) * t;
	tmp = 0.0;
	if (abs(l) <= 2e-101)
		tmp = 2.0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))));
	elseif (abs(l) <= 1.6e+142)
		tmp = 2.0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)));
	else
		tmp = 2.0 / (k * (((sin(k) * tan(k)) * (t / abs(l))) * (k / abs(l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 2e-101], N[(2.0 / N[(k * N[(k * N[(N[(t$95$1 / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 1.6e+142], N[(2.0 / N[(N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.0000000000000001e-101

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if 2.0000000000000001e-101 < l < 1.60000000000000003e142

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.3

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

   8. Applied rewrites 78.3%

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

   if 1.60000000000000003e142 < l

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 86.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.5

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

   8. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 90.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.6

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

   10. Applied rewrites 90.6%

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

Alternative 5: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 100:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{\frac{k}{\ell} \cdot k}{\ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 100.0)
   (/ 2.0 (/ (* (* (* (tan k) t) (* (sin k) k)) (/ k l)) l))
   (/ 2.0 (* (* t (* (tan k) (sin k))) (/ (* (/ k l) k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 100.0) {
		tmp = 2.0 / ((((tan(k) * t) * (sin(k) * k)) * (k / l)) / l);
	} else {
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (((k / l) * k) / l));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= 100.0d0) then
        tmp = 2.0d0 / ((((tan(k) * t) * (sin(k) * k)) * (k / l)) / l)
    else
        tmp = 2.0d0 / ((t * (tan(k) * sin(k))) * (((k / l) * k) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 100.0) {
		tmp = 2.0 / ((((Math.tan(k) * t) * (Math.sin(k) * k)) * (k / l)) / l);
	} else {
		tmp = 2.0 / ((t * (Math.tan(k) * Math.sin(k))) * (((k / l) * k) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 100.0:
		tmp = 2.0 / ((((math.tan(k) * t) * (math.sin(k) * k)) * (k / l)) / l)
	else:
		tmp = 2.0 / ((t * (math.tan(k) * math.sin(k))) * (((k / l) * k) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 100.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * t) * Float64(sin(k) * k)) * Float64(k / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(tan(k) * sin(k))) * Float64(Float64(Float64(k / l) * k) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 100.0)
		tmp = 2.0 / ((((tan(k) * t) * (sin(k) * k)) * (k / l)) / l);
	else
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (((k / l) * k) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 100.0], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 100

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 91.0

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

   8. Applied rewrites 91.0%

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

   if 100 < t

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.1

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

   8. Applied rewrites 89.1%

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

Alternative 6: 92.3% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (/ (* (* (/ k l) (* (* t (sin k)) (tan k))) k) l)))

C

double code(double t, double l, double k) {
	return 2.0 / ((((k / l) * ((t * sin(k)) * tan(k))) * k) / l);
}

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 / ((((k / l) * ((t * sin(k)) * tan(k))) * k) / l)
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / ((((k / l) * ((t * math.sin(k)) * math.tan(k))) * k) / l)

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(Float64(t * sin(k)) * tan(k))) * k) / l))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 73.8

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

4. Applied rewrites 73.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l/ N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. tan-quot N/A

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

   16. lift-tan.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lift-pow.f64 N/A

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

   19. unpow2 N/A

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

   20. associate-/l* N/A

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

6. Applied rewrites 77.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 91.0

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

8. Applied rewrites 91.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 93.3

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

10. Applied rewrites 93.3%

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

Alternative 7: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \frac{\left|k\right|}{\ell}\\ t_3 := \tan \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 3.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{2}{\frac{t\_3 \cdot \left(t \cdot \left(\left(t\_1 \cdot \left|k\right|\right) \cdot t\_2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left|k\right| \cdot \left(\left(\left(t\_1 \cdot t\_3\right) \cdot \frac{t}{\ell}\right) \cdot t\_2\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))) (t_2 (/ (fabs k) l)) (t_3 (tan (fabs k))))
   (if (<= (fabs k) 3.2e+156)
     (/ 2.0 (/ (* t_3 (* t (* (* t_1 (fabs k)) t_2))) l))
     (/ 2.0 (* (fabs k) (* (* (* t_1 t_3) (/ t l)) t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double t_2 = fabs(k) / l;
	double t_3 = tan(fabs(k));
	double tmp;
	if (fabs(k) <= 3.2e+156) {
		tmp = 2.0 / ((t_3 * (t * ((t_1 * fabs(k)) * t_2))) / l);
	} else {
		tmp = 2.0 / (fabs(k) * (((t_1 * t_3) * (t / l)) * t_2));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(abs(k))
    t_2 = abs(k) / l
    t_3 = tan(abs(k))
    if (abs(k) <= 3.2d+156) then
        tmp = 2.0d0 / ((t_3 * (t * ((t_1 * abs(k)) * t_2))) / l)
    else
        tmp = 2.0d0 / (abs(k) * (((t_1 * t_3) * (t / l)) * t_2))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double t_2 = Math.abs(k) / l;
	double t_3 = Math.tan(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 3.2e+156) {
		tmp = 2.0 / ((t_3 * (t * ((t_1 * Math.abs(k)) * t_2))) / l);
	} else {
		tmp = 2.0 / (Math.abs(k) * (((t_1 * t_3) * (t / l)) * t_2));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	t_2 = math.fabs(k) / l
	t_3 = math.tan(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 3.2e+156:
		tmp = 2.0 / ((t_3 * (t * ((t_1 * math.fabs(k)) * t_2))) / l)
	else:
		tmp = 2.0 / (math.fabs(k) * (((t_1 * t_3) * (t / l)) * t_2))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = Float64(abs(k) / l)
	t_3 = tan(abs(k))
	tmp = 0.0
	if (abs(k) <= 3.2e+156)
		tmp = Float64(2.0 / Float64(Float64(t_3 * Float64(t * Float64(Float64(t_1 * abs(k)) * t_2))) / l));
	else
		tmp = Float64(2.0 / Float64(abs(k) * Float64(Float64(Float64(t_1 * t_3) * Float64(t / l)) * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	t_2 = abs(k) / l;
	t_3 = tan(abs(k));
	tmp = 0.0;
	if (abs(k) <= 3.2e+156)
		tmp = 2.0 / ((t_3 * (t * ((t_1 * abs(k)) * t_2))) / l);
	else
		tmp = 2.0 / (abs(k) * (((t_1 * t_3) * (t / l)) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.2e+156], N[(2.0 / N[(N[(t$95$3 * N[(t * N[(N[(t$95$1 * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Abs[k], $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.20000000000000002e156

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. tan-quot N/A

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

      16. lift-tan.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/l* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 91.0

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

   8. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

   if 3.20000000000000002e156 < k

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{\color{blue}{2}}}\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   6. Applied rewrites 75.4%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\frac{t}{\ell}}{\ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      9. lower-/.f64 86.5

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

      12. lower-*.f64 86.5

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

   8. Applied rewrites 86.5%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell}}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(\frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell} \cdot \color{blue}{k}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(\frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell} \cdot k\right)} \]

      4. mult-flip N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{1}{\ell}\right) \cdot k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left(\frac{1}{\ell} \cdot k\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{1}{\ell}}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{k}{\color{blue}{\ell}}\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{k}{\color{blue}{\ell}}\right)} \]

      9. lower-*.f64 90.6

         \[\leadsto \frac{2}{k \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\frac{k}{\ell}}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{\color{blue}{k}}{\ell}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(\left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{k}}{\ell}\right)} \]

      12. lower-*.f64 90.6

          \[\leadsto \frac{2}{k \cdot \left(\left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{k}}{\ell}\right)} \]

   10. Applied rewrites 90.6%

       \[\leadsto \frac{2}{k \cdot \left(\left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\frac{k}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \tan k \cdot t\\ \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_1}{\left|\ell\right|} \cdot \frac{\sin k}{\left|\ell\right|}\right)\right)}\\ \mathbf{elif}\;\left|\ell\right| \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{k}{\left|\ell\right| \cdot \left|\ell\right|} \cdot \left(t\_1 \cdot \left(\sin k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\left|\ell\right|} \cdot \left(\sin k \cdot \tan k\right)}{\left|\ell\right|}\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) t)))
   (if (<= (fabs l) 2e-101)
     (/ 2.0 (* k (* k (* (/ t_1 (fabs l)) (/ (sin k) (fabs l))))))
     (if (<= (fabs l) 6.5e+149)
       (/ 2.0 (* (/ k (* (fabs l) (fabs l))) (* t_1 (* (sin k) k))))
       (/
        2.0
        (* k (* k (/ (* (/ t (fabs l)) (* (sin k) (tan k))) (fabs l)))))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * t;
	double tmp;
	if (fabs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / fabs(l)) * (sin(k) / fabs(l)))));
	} else if (fabs(l) <= 6.5e+149) {
		tmp = 2.0 / ((k / (fabs(l) * fabs(l))) * (t_1 * (sin(k) * k)));
	} else {
		tmp = 2.0 / (k * (k * (((t / fabs(l)) * (sin(k) * tan(k))) / fabs(l))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = tan(k) * t
    if (abs(l) <= 2d-101) then
        tmp = 2.0d0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))))
    else if (abs(l) <= 6.5d+149) then
        tmp = 2.0d0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)))
    else
        tmp = 2.0d0 / (k * (k * (((t / abs(l)) * (sin(k) * tan(k))) / abs(l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(k) * t;
	double tmp;
	if (Math.abs(l) <= 2e-101) {
		tmp = 2.0 / (k * (k * ((t_1 / Math.abs(l)) * (Math.sin(k) / Math.abs(l)))));
	} else if (Math.abs(l) <= 6.5e+149) {
		tmp = 2.0 / ((k / (Math.abs(l) * Math.abs(l))) * (t_1 * (Math.sin(k) * k)));
	} else {
		tmp = 2.0 / (k * (k * (((t / Math.abs(l)) * (Math.sin(k) * Math.tan(k))) / Math.abs(l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(k) * t
	tmp = 0
	if math.fabs(l) <= 2e-101:
		tmp = 2.0 / (k * (k * ((t_1 / math.fabs(l)) * (math.sin(k) / math.fabs(l)))))
	elif math.fabs(l) <= 6.5e+149:
		tmp = 2.0 / ((k / (math.fabs(l) * math.fabs(l))) * (t_1 * (math.sin(k) * k)))
	else:
		tmp = 2.0 / (k * (k * (((t / math.fabs(l)) * (math.sin(k) * math.tan(k))) / math.fabs(l))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * t)
	tmp = 0.0
	if (abs(l) <= 2e-101)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_1 / abs(l)) * Float64(sin(k) / abs(l))))));
	elseif (abs(l) <= 6.5e+149)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(abs(l) * abs(l))) * Float64(t_1 * Float64(sin(k) * k))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(Float64(t / abs(l)) * Float64(sin(k) * tan(k))) / abs(l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(k) * t;
	tmp = 0.0;
	if (abs(l) <= 2e-101)
		tmp = 2.0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))));
	elseif (abs(l) <= 6.5e+149)
		tmp = 2.0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)));
	else
		tmp = 2.0 / (k * (k * (((t / abs(l)) * (sin(k) * tan(k))) / abs(l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 2e-101], N[(2.0 / N[(k * N[(k * N[(N[(t$95$1 / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 6.5e+149], N[(2.0 / N[(N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.0000000000000001e-101

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{\color{blue}{2}}}\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   6. Applied rewrites 75.4%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\left(\tan k \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell} \cdot \ell}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot \left(\tan k \cdot \sin k\right)}{\ell \cdot \ell}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\left(t \cdot \tan k\right) \cdot \sin k}{\color{blue}{\ell} \cdot \ell}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\left(t \cdot \tan k\right) \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t \cdot \tan k}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t \cdot \tan k}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t \cdot \tan k}{\ell} \cdot \frac{\color{blue}{\sin k}}{\ell}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \frac{\sin \color{blue}{k}}{\ell}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \frac{\sin \color{blue}{k}}{\ell}\right)\right)} \]

      13. lower-/.f64 89.7

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \frac{\sin k}{\color{blue}{\ell}}\right)\right)} \]

   8. Applied rewrites 89.7%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}\right)\right)} \]

   if 2.0000000000000001e-101 < l < 6.50000000000000015e149

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\sin k \cdot k\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\sin k \cdot k\right)}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\color{blue}{\sin k} \cdot k\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\color{blue}{\sin k} \cdot k\right)\right)} \]

      13. lower-*.f64 78.3

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot \color{blue}{k}\right)\right)} \]

   8. Applied rewrites 78.3%

      \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right)}} \]

   if 6.50000000000000015e149 < l

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{\color{blue}{2}}}\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   6. Applied rewrites 75.4%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\frac{t}{\ell}}{\ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      9. lower-/.f64 86.5

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

      12. lower-*.f64 86.5

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

   8. Applied rewrites 86.5%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \tan k \cdot t\\ t_2 := \frac{2}{k \cdot \left(k \cdot \left(\frac{t\_1}{\left|\ell\right|} \cdot \frac{\sin k}{\left|\ell\right|}\right)\right)}\\ \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{-101}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\left|\ell\right| \leq 6.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{k}{\left|\ell\right| \cdot \left|\ell\right|} \cdot \left(t\_1 \cdot \left(\sin k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) t))
        (t_2 (/ 2.0 (* k (* k (* (/ t_1 (fabs l)) (/ (sin k) (fabs l))))))))
   (if (<= (fabs l) 2e-101)
     t_2
     (if (<= (fabs l) 6.5e+149)
       (/ 2.0 (* (/ k (* (fabs l) (fabs l))) (* t_1 (* (sin k) k))))
       t_2))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * t;
	double t_2 = 2.0 / (k * (k * ((t_1 / fabs(l)) * (sin(k) / fabs(l)))));
	double tmp;
	if (fabs(l) <= 2e-101) {
		tmp = t_2;
	} else if (fabs(l) <= 6.5e+149) {
		tmp = 2.0 / ((k / (fabs(l) * fabs(l))) * (t_1 * (sin(k) * k)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = tan(k) * t
    t_2 = 2.0d0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))))
    if (abs(l) <= 2d-101) then
        tmp = t_2
    else if (abs(l) <= 6.5d+149) then
        tmp = 2.0d0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(k) * t;
	double t_2 = 2.0 / (k * (k * ((t_1 / Math.abs(l)) * (Math.sin(k) / Math.abs(l)))));
	double tmp;
	if (Math.abs(l) <= 2e-101) {
		tmp = t_2;
	} else if (Math.abs(l) <= 6.5e+149) {
		tmp = 2.0 / ((k / (Math.abs(l) * Math.abs(l))) * (t_1 * (Math.sin(k) * k)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(k) * t
	t_2 = 2.0 / (k * (k * ((t_1 / math.fabs(l)) * (math.sin(k) / math.fabs(l)))))
	tmp = 0
	if math.fabs(l) <= 2e-101:
		tmp = t_2
	elif math.fabs(l) <= 6.5e+149:
		tmp = 2.0 / ((k / (math.fabs(l) * math.fabs(l))) * (t_1 * (math.sin(k) * k)))
	else:
		tmp = t_2
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * t)
	t_2 = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_1 / abs(l)) * Float64(sin(k) / abs(l))))))
	tmp = 0.0
	if (abs(l) <= 2e-101)
		tmp = t_2;
	elseif (abs(l) <= 6.5e+149)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(abs(l) * abs(l))) * Float64(t_1 * Float64(sin(k) * k))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(k) * t;
	t_2 = 2.0 / (k * (k * ((t_1 / abs(l)) * (sin(k) / abs(l)))));
	tmp = 0.0;
	if (abs(l) <= 2e-101)
		tmp = t_2;
	elseif (abs(l) <= 6.5e+149)
		tmp = 2.0 / ((k / (abs(l) * abs(l))) * (t_1 * (sin(k) * k)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(k * N[(k * N[(N[(t$95$1 / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 2e-101], t$95$2, If[LessEqual[N[Abs[l], $MachinePrecision], 6.5e+149], N[(2.0 / N[(N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.0000000000000001e-101 or 6.50000000000000015e149 < l

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{\color{blue}{2}}}\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   6. Applied rewrites 75.4%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\left(\tan k \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell} \cdot \ell}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot \left(\tan k \cdot \sin k\right)}{\ell \cdot \ell}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\left(t \cdot \tan k\right) \cdot \sin k}{\color{blue}{\ell} \cdot \ell}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\left(t \cdot \tan k\right) \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t \cdot \tan k}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t \cdot \tan k}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t \cdot \tan k}{\ell} \cdot \frac{\color{blue}{\sin k}}{\ell}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \frac{\sin \color{blue}{k}}{\ell}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \frac{\sin \color{blue}{k}}{\ell}\right)\right)} \]

      13. lower-/.f64 89.7

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \frac{\sin k}{\color{blue}{\ell}}\right)\right)} \]

   8. Applied rewrites 89.7%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\tan k \cdot t}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}\right)\right)} \]

   if 2.0000000000000001e-101 < l < 6.50000000000000015e149

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\sin k \cdot k\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\sin k \cdot k\right)}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\color{blue}{\sin k} \cdot k\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\color{blue}{\sin k} \cdot k\right)\right)} \]

      13. lower-*.f64 78.3

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot \color{blue}{k}\right)\right)} \]

   8. Applied rewrites 78.3%

      \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right) \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.000112:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left|k\right| \cdot t\right) \cdot t\_1\right) \cdot \frac{\left|k\right|}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left|k\right|}{\ell \cdot \ell} \cdot \left(\left(\tan \left(\left|k\right|\right) \cdot t\right) \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin (fabs k)) (fabs k))))
   (if (<= (fabs k) 0.000112)
     (/ 2.0 (/ (* (* (* (fabs k) t) t_1) (/ (fabs k) l)) l))
     (/ 2.0 (* (/ (fabs k) (* l l)) (* (* (tan (fabs k)) t) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k)) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.000112) {
		tmp = 2.0 / ((((fabs(k) * t) * t_1) * (fabs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((fabs(k) / (l * l)) * ((tan(fabs(k)) * t) * t_1));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(abs(k)) * abs(k)
    if (abs(k) <= 0.000112d0) then
        tmp = 2.0d0 / ((((abs(k) * t) * t_1) * (abs(k) / l)) / l)
    else
        tmp = 2.0d0 / ((abs(k) / (l * l)) * ((tan(abs(k)) * t) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k)) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.000112) {
		tmp = 2.0 / ((((Math.abs(k) * t) * t_1) * (Math.abs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.abs(k) / (l * l)) * ((Math.tan(Math.abs(k)) * t) * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k)) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.000112:
		tmp = 2.0 / ((((math.fabs(k) * t) * t_1) * (math.fabs(k) / l)) / l)
	else:
		tmp = 2.0 / ((math.fabs(k) / (l * l)) * ((math.tan(math.fabs(k)) * t) * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(abs(k)) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.000112)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) * t_1) * Float64(abs(k) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(k) / Float64(l * l)) * Float64(Float64(tan(abs(k)) * t) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k)) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.000112)
		tmp = 2.0 / ((((abs(k) * t) * t_1) * (abs(k) / l)) / l);
	else
		tmp = 2.0 / ((abs(k) / (l * l)) * ((tan(abs(k)) * t) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.000112], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.11999999999999998e-4

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      18. lower-/.f64 91.0

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   8. Applied rewrites 91.0%

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.1%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   if 1.11999999999999998e-4 < k

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\sin k \cdot k\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(t \cdot \tan k\right) \cdot \color{blue}{\left(\sin k \cdot k\right)}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\color{blue}{\sin k} \cdot k\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\color{blue}{\sin k} \cdot k\right)\right)} \]

      13. lower-*.f64 78.3

          \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot \color{blue}{k}\right)\right)} \]

   8. Applied rewrites 78.3%

      \[\leadsto \frac{2}{\frac{k}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right) \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.000112:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left|k\right| \cdot t\right) \cdot t\_1\right) \cdot \frac{\left|k\right|}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan \left(\left|k\right|\right) \cdot \left(\left(t\_1 \cdot t\right) \cdot \frac{\left|k\right|}{\ell \cdot \ell}\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin (fabs k)) (fabs k))))
   (if (<= (fabs k) 0.000112)
     (/ 2.0 (/ (* (* (* (fabs k) t) t_1) (/ (fabs k) l)) l))
     (/ 2.0 (* (tan (fabs k)) (* (* t_1 t) (/ (fabs k) (* l l))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k)) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.000112) {
		tmp = 2.0 / ((((fabs(k) * t) * t_1) * (fabs(k) / l)) / l);
	} else {
		tmp = 2.0 / (tan(fabs(k)) * ((t_1 * t) * (fabs(k) / (l * l))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(abs(k)) * abs(k)
    if (abs(k) <= 0.000112d0) then
        tmp = 2.0d0 / ((((abs(k) * t) * t_1) * (abs(k) / l)) / l)
    else
        tmp = 2.0d0 / (tan(abs(k)) * ((t_1 * t) * (abs(k) / (l * l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k)) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.000112) {
		tmp = 2.0 / ((((Math.abs(k) * t) * t_1) * (Math.abs(k) / l)) / l);
	} else {
		tmp = 2.0 / (Math.tan(Math.abs(k)) * ((t_1 * t) * (Math.abs(k) / (l * l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k)) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.000112:
		tmp = 2.0 / ((((math.fabs(k) * t) * t_1) * (math.fabs(k) / l)) / l)
	else:
		tmp = 2.0 / (math.tan(math.fabs(k)) * ((t_1 * t) * (math.fabs(k) / (l * l))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(abs(k)) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.000112)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) * t_1) * Float64(abs(k) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(tan(abs(k)) * Float64(Float64(t_1 * t) * Float64(abs(k) / Float64(l * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k)) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.000112)
		tmp = 2.0 / ((((abs(k) * t) * t_1) * (abs(k) / l)) / l);
	else
		tmp = 2.0 / (tan(abs(k)) * ((t_1 * t) * (abs(k) / (l * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.000112], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 * t), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.11999999999999998e-4

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      18. lower-/.f64 91.0

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   8. Applied rewrites 91.0%

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.1%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   if 1.11999999999999998e-4 < k

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      18. lower-/.f64 91.0

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   8. Applied rewrites 91.0%

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{k}{\ell}}{\ell}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{k}{\ell}}}{\ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{k}}{\ell}}{\ell}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(t \cdot \left(\sin k \cdot k\right)\right)\right) \cdot \frac{\color{blue}{\frac{k}{\ell}}}{\ell}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(t \cdot \left(\sin k \cdot k\right)\right)\right) \cdot \frac{\frac{k}{\ell}}{\ell}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(t \cdot \left(\sin k \cdot k\right)\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(t \cdot \left(\sin k \cdot k\right)\right)\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\tan k \cdot \left(t \cdot \left(\sin k \cdot k\right)\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\left(\left(t \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\left(\left(t \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\tan k \cdot \left(\left(t \cdot \left(\sin k \cdot k\right)\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\tan k \cdot \left(\left(\left(\sin k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{k}}{\ell \cdot \ell}\right)} \]

      15. lower-*.f64 78.6

          \[\leadsto \frac{2}{\tan k \cdot \left(\left(\left(\sin k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{k}}{\ell \cdot \ell}\right)} \]

   10. Applied rewrites 78.6%

       \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\left(\left(\left(\sin k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 83.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right) \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.0014:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left|k\right| \cdot \left(t + 0.3333333333333333 \cdot \left({\left(\left|k\right|\right)}^{2} \cdot t\right)\right)\right) \cdot t\_1\right) \cdot \frac{\left|k\right|}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_1 \cdot \left(\left(\tan \left(\left|k\right|\right) \cdot t\right) \cdot \left|k\right|\right)}{\ell \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin (fabs k)) (fabs k))))
   (if (<= (fabs k) 0.0014)
     (/
      2.0
      (/
       (*
        (*
         (* (fabs k) (+ t (* 0.3333333333333333 (* (pow (fabs k) 2.0) t))))
         t_1)
        (/ (fabs k) l))
       l))
     (/ 2.0 (/ (* t_1 (* (* (tan (fabs k)) t) (fabs k))) (* l l))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k)) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.0014) {
		tmp = 2.0 / ((((fabs(k) * (t + (0.3333333333333333 * (pow(fabs(k), 2.0) * t)))) * t_1) * (fabs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((t_1 * ((tan(fabs(k)) * t) * fabs(k))) / (l * l));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(abs(k)) * abs(k)
    if (abs(k) <= 0.0014d0) then
        tmp = 2.0d0 / ((((abs(k) * (t + (0.3333333333333333d0 * ((abs(k) ** 2.0d0) * t)))) * t_1) * (abs(k) / l)) / l)
    else
        tmp = 2.0d0 / ((t_1 * ((tan(abs(k)) * t) * abs(k))) / (l * l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k)) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.0014) {
		tmp = 2.0 / ((((Math.abs(k) * (t + (0.3333333333333333 * (Math.pow(Math.abs(k), 2.0) * t)))) * t_1) * (Math.abs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((t_1 * ((Math.tan(Math.abs(k)) * t) * Math.abs(k))) / (l * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k)) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.0014:
		tmp = 2.0 / ((((math.fabs(k) * (t + (0.3333333333333333 * (math.pow(math.fabs(k), 2.0) * t)))) * t_1) * (math.fabs(k) / l)) / l)
	else:
		tmp = 2.0 / ((t_1 * ((math.tan(math.fabs(k)) * t) * math.fabs(k))) / (l * l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(abs(k)) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.0014)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * Float64(t + Float64(0.3333333333333333 * Float64((abs(k) ^ 2.0) * t)))) * t_1) * Float64(abs(k) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(Float64(tan(abs(k)) * t) * abs(k))) / Float64(l * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k)) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.0014)
		tmp = 2.0 / ((((abs(k) * (t + (0.3333333333333333 * ((abs(k) ^ 2.0) * t)))) * t_1) * (abs(k) / l)) / l);
	else
		tmp = 2.0 / ((t_1 * ((tan(abs(k)) * t) * abs(k))) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.0014], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * N[(t + N[(0.3333333333333333 * N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00139999999999999999

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      18. lower-/.f64 91.0

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   8. Applied rewrites 91.0%

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(t + \frac{1}{3} \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(t + \frac{1}{3} \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(t + \frac{1}{3} \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(t + \frac{1}{3} \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(t + \frac{1}{3} \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

       5. lower-pow.f64 71.6

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(t + 0.3333333333333333 \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   11. Applied rewrites 71.6%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot \left(t + 0.3333333333333333 \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   if 0.00139999999999999999 < k

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      18. lower-/.f64 91.0

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   8. Applied rewrites 91.0%

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k}{\ell}}{\ell}} \]

      5. associate-/l/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k}{\ell \cdot \color{blue}{\ell}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k}{\ell \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot k\right) \cdot \left(\tan k \cdot t\right)\right) \cdot k}{\ell \cdot \ell}} \]

      10. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\sin k \cdot k\right) \cdot \left(\left(\tan k \cdot t\right) \cdot k\right)}{\color{blue}{\ell} \cdot \ell}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\sin k \cdot k\right) \cdot \left(\left(\tan k \cdot t\right) \cdot k\right)}{\color{blue}{\ell} \cdot \ell}} \]

      12. lower-*.f64 76.1

          \[\leadsto \frac{2}{\frac{\left(\sin k \cdot k\right) \cdot \left(\left(\tan k \cdot t\right) \cdot k\right)}{\ell \cdot \ell}} \]

   10. Applied rewrites 76.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot k\right) \cdot \left(\left(\tan k \cdot t\right) \cdot k\right)}{\ell \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.6% accurate, 1.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\frac{\left|t\right|}{\ell} \cdot \left(\sin k \cdot k\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \left|t\right|\right) \cdot {k}^{2}\right) \cdot \frac{k}{\ell}}{\ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 6e-83)
    (/ 2.0 (* k (* k (/ (* (/ (fabs t) l) (* (sin k) k)) l))))
    (/ 2.0 (/ (* (* (* (tan k) (fabs t)) (pow k 2.0)) (/ k l)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 6e-83) {
		tmp = 2.0 / (k * (k * (((fabs(t) / l) * (sin(k) * k)) / l)));
	} else {
		tmp = 2.0 / ((((tan(k) * fabs(t)) * pow(k, 2.0)) * (k / l)) / l);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 6e-83) {
		tmp = 2.0 / (k * (k * (((Math.abs(t) / l) * (Math.sin(k) * k)) / l)));
	} else {
		tmp = 2.0 / ((((Math.tan(k) * Math.abs(t)) * Math.pow(k, 2.0)) * (k / l)) / l);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 6e-83:
		tmp = 2.0 / (k * (k * (((math.fabs(t) / l) * (math.sin(k) * k)) / l)))
	else:
		tmp = 2.0 / ((((math.tan(k) * math.fabs(t)) * math.pow(k, 2.0)) * (k / l)) / l)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 6e-83)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(Float64(abs(t) / l) * Float64(sin(k) * k)) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * abs(t)) * (k ^ 2.0)) * Float64(k / l)) / l));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 6e-83)
		tmp = 2.0 / (k * (k * (((abs(t) / l) * (sin(k) * k)) / l)));
	else
		tmp = 2.0 / ((((tan(k) * abs(t)) * (k ^ 2.0)) * (k / l)) / l);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 6e-83], N[(2.0 / N[(k * N[(k * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.00000000000000021e-83

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{\color{blue}{2}}}\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   6. Applied rewrites 75.4%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\frac{t}{\ell}}{\ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      9. lower-/.f64 86.5

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

      12. lower-*.f64 86.5

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

   8. Applied rewrites 86.5%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell}}\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot k\right)}{\ell}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.7%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot k\right)}{\ell}\right)} \]

   if 6.00000000000000021e-83 < t

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      18. lower-/.f64 91.0

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   8. Applied rewrites 91.0%

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot {k}^{2}\right) \cdot \frac{k}{\ell}}{\ell}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 72.1

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot {k}^{2}\right) \cdot \frac{k}{\ell}}{\ell}} \]

   11. Applied rewrites 72.1%

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot {k}^{2}\right) \cdot \frac{k}{\ell}}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{+98}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\frac{\left|t\right|}{\ell} \cdot t\_1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot \left|t\right|\right) \cdot t\_1\right) \cdot \frac{k}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) k)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1e+98)
      (/ 2.0 (* k (* k (/ (* (/ (fabs t) l) t_1) l))))
      (/ 2.0 (/ (* (* (* k (fabs t)) t_1) (/ k l)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * k;
	double tmp;
	if (fabs(t) <= 1e+98) {
		tmp = 2.0 / (k * (k * (((fabs(t) / l) * t_1) / l)));
	} else {
		tmp = 2.0 / ((((k * fabs(t)) * t_1) * (k / l)) / l);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * k;
	double tmp;
	if (Math.abs(t) <= 1e+98) {
		tmp = 2.0 / (k * (k * (((Math.abs(t) / l) * t_1) / l)));
	} else {
		tmp = 2.0 / ((((k * Math.abs(t)) * t_1) * (k / l)) / l);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * k
	tmp = 0
	if math.fabs(t) <= 1e+98:
		tmp = 2.0 / (k * (k * (((math.fabs(t) / l) * t_1) / l)))
	else:
		tmp = 2.0 / ((((k * math.fabs(t)) * t_1) * (k / l)) / l)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * k)
	tmp = 0.0
	if (abs(t) <= 1e+98)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(Float64(abs(t) / l) * t_1) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * abs(t)) * t_1) * Float64(k / l)) / l));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * k;
	tmp = 0.0;
	if (abs(t) <= 1e+98)
		tmp = 2.0 / (k * (k * (((abs(t) / l) * t_1) / l)));
	else
		tmp = 2.0 / ((((k * abs(t)) * t_1) * (k / l)) / l);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e+98], N[(2.0 / N[(k * N[(k * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999998e97

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{\color{blue}{2}}}\right)\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   6. Applied rewrites 75.4%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\frac{t}{\ell}}{\ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      9. lower-/.f64 86.5

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

      12. lower-*.f64 86.5

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

   8. Applied rewrites 86.5%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell}}\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot k\right)}{\ell}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.7%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot k\right)}{\ell}\right)} \]

   if 9.99999999999999998e97 < t

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.8

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      15. tan-quot N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      16. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

      20. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

      18. lower-/.f64 91.0

          \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   8. Applied rewrites 91.0%

      \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.1%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 72.1% accurate, 2.3× speedup

Math

\[\frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (/ (* (* (* k t) (* (sin k) k)) (/ k l)) l)))

C

double code(double t, double l, double k) {
	return 2.0 / ((((k * t) * (sin(k) * k)) * (k / l)) / l);
}

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 / ((((k * t) * (sin(k) * k)) * (k / l)) / l)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((k * t) * (Math.sin(k) * k)) * (k / l)) / l);
}

Python

def code(t, l, k):
	return 2.0 / ((((k * t) * (math.sin(k) * k)) * (k / l)) / l)

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64(k * t) * Float64(sin(k) * k)) * Float64(k / l)) / l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((((k * t) * (sin(k) * k)) * (k / l)) / l);
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

   9. lower-cos.f64 73.8

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

4. Applied rewrites 73.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   4. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   12. associate-*l/ N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   13. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   14. lift-cos.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   15. tan-quot N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   16. lift-tan.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   18. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   19. unpow2 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

   20. associate-/l* N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

6. Applied rewrites 77.4%

   \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   13. associate-*l* N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   15. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   18. lower-/.f64 91.0

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

8. Applied rewrites 91.0%

   \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]
10. Step-by-step derivation

11. Applied rewrites 72.1%

    \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]
12. Add Preprocessing

Alternative 16: 72.0% accurate, 3.5× speedup

Math

\[\frac{2}{k \cdot \left(k \cdot \frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}\right)} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* k (* k (/ (/ (* (pow k 2.0) t) l) l)))))

C

double code(double t, double l, double k) {
	return 2.0 / (k * (k * (((pow(k, 2.0) * t) / l) / l)));
}

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 / (k * (k * ((((k ** 2.0d0) * t) / l) / l)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (k * (k * (((Math.pow(k, 2.0) * t) / l) / l)));
}

Python

def code(t, l, k):
	return 2.0 / (k * (k * (((math.pow(k, 2.0) * t) / l) / l)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(k * Float64(k * Float64(Float64(Float64((k ^ 2.0) * t) / l) / l))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (k * (k * ((((k ^ 2.0) * t) / l) / l)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(k * N[(k * N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

   9. lower-cos.f64 73.8

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

4. Applied rewrites 73.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{{\ell}^{2}}}\right)} \]

   13. times-frac N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{\color{blue}{2}}}\right)\right)} \]

   15. pow2 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}\right)\right)} \]

   17. lift-/.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)\right)} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

6. Applied rewrites 75.4%

   \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \]

   5. associate-/r* N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{\frac{t}{\ell}}{\ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)\right)\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\color{blue}{\ell}}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

   9. lower-/.f64 86.5

      \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell}\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

   12. lower-*.f64 86.5

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}\right)} \]

8. Applied rewrites 86.5%

   \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell}}\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}\right)} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}\right)} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}\right)} \]

    3. lower-pow.f64 72.0

       \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}\right)} \]

11. Applied rewrites 72.0%

    \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}\right)} \]
12. Add Preprocessing

Alternative 17: 70.2% accurate, 3.9× speedup

Math

\[\frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (/ (* (* (pow k 3.0) t) (/ k l)) l)))

C

double code(double t, double l, double k) {
	return 2.0 / (((pow(k, 3.0) * t) * (k / l)) / l);
}

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 / ((((k ** 3.0d0) * t) * (k / l)) / l)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((Math.pow(k, 3.0) * t) * (k / l)) / l);
}

Python

def code(t, l, k):
	return 2.0 / (((math.pow(k, 3.0) * t) * (k / l)) / l)

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64((k ^ 3.0) * t) * Float64(k / l)) / l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((((k ^ 3.0) * t) * (k / l)) / l);
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[Power[k, 3.0], $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

   9. lower-cos.f64 73.8

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

4. Applied rewrites 73.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   4. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   12. associate-*l/ N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   13. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   14. lift-cos.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{\cos k} \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   15. tan-quot N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   16. lift-tan.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   18. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   19. unpow2 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k \cdot k}{{\color{blue}{\ell}}^{2}}} \]

   20. associate-/l* N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right)} \]

6. Applied rewrites 77.4%

   \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\left(t \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\ell}} \]

   13. associate-*l* N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   15. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

   18. lower-/.f64 91.0

       \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\ell}} \]

8. Applied rewrites 91.0%

   \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]

    2. lower-pow.f64 70.2

       \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]

11. Applied rewrites 70.2%

    \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]
12. Add Preprocessing

Alternative 18: 69.1% accurate, 4.4× speedup

Math

\[\frac{\ell \cdot {k}^{-4}}{t} \cdot \left(\ell + \ell\right) \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (* l (pow k -4.0)) t) (+ l l)))

C

double code(double t, double l, double k) {
	return ((l * pow(k, -4.0)) / t) * (l + l);
}

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 = ((l * (k ** (-4.0d0))) / t) * (l + l)
end function

Java

public static double code(double t, double l, double k) {
	return ((l * Math.pow(k, -4.0)) / t) * (l + l);
}

Python

def code(t, l, k):
	return ((l * math.pow(k, -4.0)) / t) * (l + l)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l * (k ^ -4.0)) / t) * Float64(l + l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l * (k ^ -4.0)) / t) * (l + l);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.3

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 62.3

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 68.5

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 68.5%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{4} \cdot t}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}}\right) \]

   5. associate-*r/ N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{4} \cdot t}} \]

   6. pow2 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   7. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   8. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   9. exp-to-pow N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   10. lift-log.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   12. lift-exp.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   13. lift-/.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{e^{\log k \cdot 4} \cdot t}} \]

   14. count-2-rev N/A

       \[\leadsto \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} + \color{blue}{\frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t}} \]

8. Applied rewrites 68.5%

   \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell + \ell\right)} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\color{blue}{\ell} + \ell\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\ell}{{k}^{4}}}{t} \cdot \left(\color{blue}{\ell} + \ell\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\ell}{{k}^{4}}}{t} \cdot \left(\color{blue}{\ell} + \ell\right) \]

   5. mult-flip N/A

      \[\leadsto \frac{\ell \cdot \frac{1}{{k}^{4}}}{t} \cdot \left(\ell + \ell\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \frac{1}{{k}^{4}}}{t} \cdot \left(\ell + \ell\right) \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \frac{1}{{k}^{4}}}{t} \cdot \left(\ell + \ell\right) \]

   8. pow-flip N/A

      \[\leadsto \frac{\ell \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \left(\ell + \ell\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \left(\ell + \ell\right) \]

   10. metadata-eval 69.1

       \[\leadsto \frac{\ell \cdot {k}^{-4}}{t} \cdot \left(\ell + \ell\right) \]

10. Applied rewrites 69.1%

    \[\leadsto \frac{\ell \cdot {k}^{-4}}{t} \cdot \left(\color{blue}{\ell} + \ell\right) \]
11. Add Preprocessing

Alternative 19: 68.5% accurate, 4.4× speedup

Math

\[\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (pow k 4.0) t)) (+ l l)))

C

double code(double t, double l, double k) {
	return (l / (pow(k, 4.0) * t)) * (l + l);
}

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 = (l / ((k ** 4.0d0) * t)) * (l + l)
end function

Java

public static double code(double t, double l, double k) {
	return (l / (Math.pow(k, 4.0) * t)) * (l + l);
}

Python

def code(t, l, k):
	return (l / (math.pow(k, 4.0) * t)) * (l + l)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64((k ^ 4.0) * t)) * Float64(l + l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k ^ 4.0) * t)) * (l + l);
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.3

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 62.3

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 68.5

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 68.5%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{4} \cdot t}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}}\right) \]

   5. associate-*r/ N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{4} \cdot t}} \]

   6. pow2 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   7. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   8. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   9. exp-to-pow N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   10. lift-log.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   12. lift-exp.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   13. lift-/.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{e^{\log k \cdot 4} \cdot t}} \]

   14. count-2-rev N/A

       \[\leadsto \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} + \color{blue}{\frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t}} \]

8. Applied rewrites 68.5%

   \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell + \ell\right)} \]
9. Add Preprocessing

Alternative 20: 68.3% accurate, 4.4× speedup

Math

\[\left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

FPCore

(FPCore (t l k) :precision binary64 (* (* (/ (pow k -4.0) t) l) (+ l l)))

C

double code(double t, double l, double k) {
	return ((pow(k, -4.0) / t) * l) * (l + l);
}

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 = (((k ** (-4.0d0)) / t) * l) * (l + l)
end function

Java

public static double code(double t, double l, double k) {
	return ((Math.pow(k, -4.0) / t) * l) * (l + l);
}

Python

def code(t, l, k):
	return ((math.pow(k, -4.0) / t) * l) * (l + l)

Julia

function code(t, l, k)
	return Float64(Float64(Float64((k ^ -4.0) / t) * l) * Float64(l + l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (((k ^ -4.0) / t) * l) * (l + l);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(N[Power[k, -4.0], $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.3

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 62.3

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 68.5

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 68.5%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{4} \cdot t}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}}\right) \]

   5. associate-*r/ N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{4} \cdot t}} \]

   6. pow2 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   7. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   8. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   9. exp-to-pow N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   10. lift-log.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   12. lift-exp.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} \]

   13. lift-/.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{e^{\log k \cdot 4} \cdot t}} \]

   14. count-2-rev N/A

       \[\leadsto \frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t} + \color{blue}{\frac{{\ell}^{2}}{e^{\log k \cdot 4} \cdot t}} \]

8. Applied rewrites 68.5%

   \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell + \ell\right)} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\color{blue}{\ell} + \ell\right) \]

   2. mult-flip N/A

      \[\leadsto \left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right) \cdot \left(\color{blue}{\ell} + \ell\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{1}{{k}^{4} \cdot t} \cdot \ell\right) \cdot \left(\color{blue}{\ell} + \ell\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{{k}^{4} \cdot t} \cdot \ell\right) \cdot \left(\color{blue}{\ell} + \ell\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{{k}^{4} \cdot t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

   6. associate-/r* N/A

      \[\leadsto \left(\frac{\frac{1}{{k}^{4}}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\frac{\frac{1}{{k}^{4}}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \left(\frac{\frac{1}{{k}^{4}}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

   9. pow-flip N/A

      \[\leadsto \left(\frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto \left(\frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

   11. metadata-eval 68.3

       \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

10. Applied rewrites 68.3%

    \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\color{blue}{\ell} + \ell\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025178 
(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))))