Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 93.4%

Time: 17.4s

Alternatives: 18

Speedup: 13.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;l\_m \leq 3.3 \cdot 10^{+114}:\\ \;\;\;\;\frac{2}{\frac{t\_1 \cdot \left(\left(k \cdot t\right) \cdot \frac{k}{l\_m}\right)}{\cos k \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot \frac{t}{l\_m}\right) \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_1}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= l_m 3.3e+114)
     (/ 2.0 (/ (* t_1 (* (* k t) (/ k l_m))) (* (cos k) l_m)))
     (/ 2.0 (* (* (* k (/ t l_m)) (/ k l_m)) (/ t_1 (cos k)))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (l_m <= 3.3e+114) {
		tmp = 2.0 / ((t_1 * ((k * t) * (k / l_m))) / (cos(k) * l_m));
	} else {
		tmp = 2.0 / (((k * (t / l_m)) * (k / l_m)) * (t_1 / cos(k)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (l_m <= 3.3d+114) then
        tmp = 2.0d0 / ((t_1 * ((k * t) * (k / l_m))) / (cos(k) * l_m))
    else
        tmp = 2.0d0 / (((k * (t / l_m)) * (k / l_m)) * (t_1 / cos(k)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (l_m <= 3.3e+114) {
		tmp = 2.0 / ((t_1 * ((k * t) * (k / l_m))) / (Math.cos(k) * l_m));
	} else {
		tmp = 2.0 / (((k * (t / l_m)) * (k / l_m)) * (t_1 / Math.cos(k)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if l_m <= 3.3e+114:
		tmp = 2.0 / ((t_1 * ((k * t) * (k / l_m))) / (math.cos(k) * l_m))
	else:
		tmp = 2.0 / (((k * (t / l_m)) * (k / l_m)) * (t_1 / math.cos(k)))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (l_m <= 3.3e+114)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(Float64(k * t) * Float64(k / l_m))) / Float64(cos(k) * l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t / l_m)) * Float64(k / l_m)) * Float64(t_1 / cos(k))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (l_m <= 3.3e+114)
		tmp = 2.0 / ((t_1 * ((k * t) * (k / l_m))) / (cos(k) * l_m));
	else
		tmp = 2.0 / (((k * (t / l_m)) * (k / l_m)) * (t_1 / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l$95$m, 3.3e+114], N[(2.0 / N[(N[(t$95$1 * N[(N[(k * t), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.3000000000000001e114

   1. Initial program 33.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 82.3

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 94.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. lift-sin.f64 N/A

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

       9. lift-cos.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*l/ N/A

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

       12. frac-times N/A

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

       13. lower-/.f64 N/A

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

   11. Applied rewrites 93.3%

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

   if 3.3000000000000001e114 < l

   1. Initial program 25.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 59.3

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 89.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 94.8

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

   11. Applied rewrites 94.8%

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

Alternative 2: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;l\_m \leq 4.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\frac{t\_1 \cdot \left(\left(k \cdot t\right) \cdot \frac{k}{l\_m}\right)}{\cos k \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{ratio\_of\_squares}\left(k, l\_m\right) \cdot \left(t \cdot \frac{t\_1}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= l_m 4.8e+52)
     (/ 2.0 (/ (* t_1 (* (* k t) (/ k l_m))) (* (cos k) l_m)))
     (/ 2.0 (* (ratio-of-squares k l_m) (* t (/ t_1 (cos k))))))))

Derivation

1. Split input into 2 regimes

2. if l < 4.8e52

   1. Initial program 33.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 81.5

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 93.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. lift-sin.f64 N/A

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

       9. lift-cos.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*l/ N/A

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

       12. frac-times N/A

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

       13. lower-/.f64 N/A

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

   11. Applied rewrites 92.9%

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

   if 4.8e52 < l

   1. Initial program 27.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 66.0

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 95.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \]

      11. lift-pow.f64 N/A

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

      12. lift-cos.f64 95.2

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \]

   9. Applied rewrites 95.2%

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

Alternative 3: 90.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\sin k}^{2} \cdot t\\ \mathbf{if}\;k \leq 1.12 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{l\_m} \cdot t\_1}{l\_m \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot \frac{\cos k}{t\_1}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (pow (sin k) 2.0) t)))
   (if (<= k 1.12e+96)
     (/ 2.0 (/ (* (/ (* k k) l_m) t_1) (* l_m (cos k))))
     (* (* (ratio-of-squares l_m k) (/ (cos k) t_1)) 2.0))))

Derivation

1. Split input into 2 regimes

2. if k < 1.1199999999999999e96

   1. Initial program 32.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 82.9

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-cos.f64 90.4

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

   9. Applied rewrites 90.4%

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

   if 1.1199999999999999e96 < k

   1. Initial program 26.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 49.9

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lift-ratio-of-squares.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-ratio-of-squares.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

   8. Applied rewrites 90.7%

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

Alternative 4: 82.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{k \cdot t}{l\_m} \cdot \frac{k}{l\_m}\\ \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* (/ (* k t) l_m) (/ k l_m))))
   (if (<= k 0.0025)
     (/ 2.0 (* t_1 (* (+ (* 0.16666666666666666 (* k k)) 1.0) (* k k))))
     (/ 2.0 (* t_1 (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) (cos k)))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = ((k * t) / l_m) * (k / l_m);
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / (t_1 * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)));
	} else {
		tmp = 2.0 / (t_1 * ((0.5 - (0.5 * cos((2.0 * k)))) / cos(k)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((k * t) / l_m) * (k / l_m)
    if (k <= 0.0025d0) then
        tmp = 2.0d0 / (t_1 * (((0.16666666666666666d0 * (k * k)) + 1.0d0) * (k * k)))
    else
        tmp = 2.0d0 / (t_1 * ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) / cos(k)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = ((k * t) / l_m) * (k / l_m);
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / (t_1 * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)));
	} else {
		tmp = 2.0 / (t_1 * ((0.5 - (0.5 * Math.cos((2.0 * k)))) / Math.cos(k)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = ((k * t) / l_m) * (k / l_m)
	tmp = 0
	if k <= 0.0025:
		tmp = 2.0 / (t_1 * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)))
	else:
		tmp = 2.0 / (t_1 * ((0.5 - (0.5 * math.cos((2.0 * k)))) / math.cos(k)))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(Float64(Float64(k * t) / l_m) * Float64(k / l_m))
	tmp = 0.0
	if (k <= 0.0025)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(Float64(0.16666666666666666 * Float64(k * k)) + 1.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / cos(k))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = ((k * t) / l_m) * (k / l_m);
	tmp = 0.0;
	if (k <= 0.0025)
		tmp = 2.0 / (t_1 * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)));
	else
		tmp = 2.0 / (t_1 * ((0.5 - (0.5 * cos((2.0 * k)))) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(N[(N[(k * t), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.0025], N[(2.0 / N[(t$95$1 * N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 83.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

       9. lower-*.f64 81.4

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

   12. Applied rewrites 81.4%

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.00250000000000000005 < k

   1. Initial program 25.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 58.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 91.0%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 90.3

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

   11. Applied rewrites 90.3%

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

Alternative 5: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00044:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, l\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{ratio\_of\_squares}\left(k, l\_m\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 0.00044)
   (/
    2.0
    (*
     (* (/ (* k t) l_m) (/ k l_m))
     (* (+ (* 0.16666666666666666 (* k k)) 1.0) (* k k))))
   (if (<= k 1.02e+148)
     (/ 2.0 (* (* k k) (* (/ t (cos k)) (ratio-of-squares (sin k) l_m))))
     (/ 2.0 (* (ratio-of-squares k l_m) (/ (* (pow (sin k) 2.0) t) 1.0))))))

Derivation

1. Split input into 3 regimes

2. if k < 4.40000000000000016e-4

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 83.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

       9. lower-*.f64 81.4

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

   12. Applied rewrites 81.4%

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 4.40000000000000016e-4 < k < 1.02e148

   1. Initial program 21.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 69.0

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      6. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      7. lift-ratio-of-squares.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

   9. Applied rewrites 76.2%

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

   if 1.02e148 < k

   1. Initial program 28.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 50.7

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.0%

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

Alternative 6: 82.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 2.1:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{ratio\_of\_squares}\left(k, l\_m\right) \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 2.1)
   (/
    2.0
    (*
     (* (/ (* k t) l_m) (/ k l_m))
     (* (+ (* 0.16666666666666666 (* k k)) 1.0) (* k k))))
   (/
    2.0
    (*
     (ratio-of-squares k l_m)
     (/ (* (- 0.5 (* 0.5 (cos (+ k k)))) t) (cos k))))))

Derivation

1. Split input into 2 regimes

2. if k < 2.10000000000000009

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 83.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

       9. lower-*.f64 81.4

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

   12. Applied rewrites 81.4%

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 2.10000000000000009 < k

   1. Initial program 25.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 58.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 88.3%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(\sin k \cdot \sin k\right) \cdot t}{\cos k}} \]

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)\right) \cdot t}{\cos k}} \]

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 87.6

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

   9. Applied rewrites 87.6%

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

Alternative 7: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00032:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \sin k\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{ratio\_of\_squares}\left(k, l\_m\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 0.00032)
   (/
    2.0
    (*
     (* (/ (* k t) l_m) (/ k l_m))
     (* (+ (* 0.16666666666666666 (* k k)) 1.0) (* k k))))
   (if (<= k 1.02e+148)
     (* (* (/ (cos k) (* (* k k) t)) (ratio-of-squares l_m (sin k))) 2.0)
     (/ 2.0 (* (ratio-of-squares k l_m) (/ (* (pow (sin k) 2.0) t) 1.0))))))

Derivation

1. Split input into 3 regimes

2. if k < 3.20000000000000026e-4

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 83.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

       9. lower-*.f64 81.4

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

   12. Applied rewrites 81.4%

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 3.20000000000000026e-4 < k < 1.02e148

   1. Initial program 21.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.8%

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

   if 1.02e148 < k

   1. Initial program 28.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 50.7

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.0%

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

Alternative 8: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00032:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \sin k\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{ratio\_of\_squares}\left(k, l\_m\right) \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 0.00032)
   (/
    2.0
    (*
     (* (/ (* k t) l_m) (/ k l_m))
     (* (+ (* 0.16666666666666666 (* k k)) 1.0) (* k k))))
   (if (<= k 1.02e+148)
     (* (* (/ (cos k) (* (* k k) t)) (ratio-of-squares l_m (sin k))) 2.0)
     (/
      2.0
      (*
       (ratio-of-squares k l_m)
       (/ (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t) 1.0))))))

Derivation

1. Split input into 3 regimes

2. if k < 3.20000000000000026e-4

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 83.6

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-ratio-of-squares.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

       9. lower-*.f64 81.4

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

   12. Applied rewrites 81.4%

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 3.20000000000000026e-4 < k < 1.02e148

   1. Initial program 21.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.8%

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

   if 1.02e148 < k

   1. Initial program 28.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 50.7

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 50.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      8. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      17. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      18. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

   7. Applied rewrites 87.4%

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.0%

       \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}} \]
   11. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}} \]

       2. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{{\sin k}^{2} \cdot t}{1}} \]

       3. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(\sin k \cdot \sin k\right) \cdot t}{1}} \]

       4. sqr-sin-a N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{1}} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{1}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{1}} \]

       7. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{1}} \]

       8. lower-*.f64 58.0

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{1}} \]

   12. Applied rewrites 58.0%

       \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, l\_m\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ 2.0 (* (/ (* k (* k t)) (cos k)) (ratio-of-squares (sin k) l_m))))

Derivation

1. Initial program 31.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{{\color{blue}{\ell}}^{2}}} \]

   11. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \color{blue}{\ell}}} \]

   12. lower-ratio-of-squares.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \color{blue}{\ell}\right)} \]

   13. lift-sin.f64 77.3

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

5. Applied rewrites 77.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]

   5. lower-*.f64 81.3

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

7. Applied rewrites 81.3%

   \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]
8. Add Preprocessing

Alternative 10: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{\left(k \cdot \frac{k \cdot t}{\cos k}\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, l\_m\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ 2.0 (* (* k (/ (* k t) (cos k))) (ratio-of-squares (sin k) l_m))))

Derivation

1. Initial program 31.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{{\color{blue}{\ell}}^{2}}} \]

   11. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \color{blue}{\ell}}} \]

   12. lower-ratio-of-squares.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \color{blue}{\ell}\right)} \]

   13. lift-sin.f64 77.3

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

5. Applied rewrites 77.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]

   5. lower-*.f64 81.3

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

7. Applied rewrites 81.3%

   \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin \color{blue}{k}, \ell\right)} \]

   4. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. associate-/l* N/A

      \[\leadsto \frac{2}{\left(k \cdot \frac{k \cdot t}{\cos k}\right) \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot \frac{k \cdot t}{\cos k}\right) \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot \frac{k \cdot t}{\cos k}\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot \frac{k \cdot t}{\cos k}\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   9. lift-cos.f64 81.3

      \[\leadsto \frac{2}{\left(k \cdot \frac{k \cdot t}{\cos k}\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

9. Applied rewrites 81.3%

   \[\leadsto \frac{2}{\left(k \cdot \frac{k \cdot t}{\cos k}\right) \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]
10. Add Preprocessing

Alternative 11: 73.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{l\_m}{k} \cdot \frac{\frac{l\_m}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1.32e+16)
   (/
    2.0
    (*
     (* (/ (* k t) l_m) (/ k l_m))
     (* (+ (* 0.16666666666666666 (* k k)) 1.0) (* k k))))
   (* (* (/ l_m k) (/ (/ l_m k) t)) -0.3333333333333333)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 1.32e+16) {
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)));
	} else {
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.32d+16) then
        tmp = 2.0d0 / ((((k * t) / l_m) * (k / l_m)) * (((0.16666666666666666d0 * (k * k)) + 1.0d0) * (k * k)))
    else
        tmp = ((l_m / k) * ((l_m / k) / t)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 1.32e+16) {
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)));
	} else {
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 1.32e+16:
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)))
	else:
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 1.32e+16)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) / l_m) * Float64(k / l_m)) * Float64(Float64(Float64(0.16666666666666666 * Float64(k * k)) + 1.0) * Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(l_m / k) * Float64(Float64(l_m / k) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 1.32e+16)
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (((0.16666666666666666 * (k * k)) + 1.0) * (k * k)));
	else
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 1.32e+16], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(N[(l$95$m / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.32e16

   1. Initial program 33.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{{\color{blue}{\ell}}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \color{blue}{\ell}}} \]

      12. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \color{blue}{\ell}\right)} \]

      13. lift-sin.f64 84.0

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 84.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      8. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      17. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      18. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

   7. Applied rewrites 90.6%

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

      2. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{\cos k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2}} \cdot t}{\cos k}} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot \color{blue}{t}}{\cos k}} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   9. Applied rewrites 94.0%

      \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot {k}^{2}\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

       9. lower-*.f64 80.8

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \left(k \cdot k\right)\right)} \]

   12. Applied rewrites 80.8%

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(0.16666666666666666 \cdot \left(k \cdot k\right) + 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 1.32e16 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      7. lower-ratio-of-squares.f64 48.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
   9. Step-by-step derivation

      1. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      5. lower-/.f64 48.6

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 48.6%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       3. associate-/l* N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       5. lower-/.f64 49.1

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]

   12. Applied rewrites 49.1%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{l\_m}{k} \cdot \frac{\frac{l\_m}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1.32e+16)
   (/ 2.0 (* (* (/ (* k t) l_m) (/ k l_m)) (* k k)))
   (* (* (/ l_m k) (/ (/ l_m k) t)) -0.3333333333333333)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 1.32e+16) {
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (k * k));
	} else {
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.32d+16) then
        tmp = 2.0d0 / ((((k * t) / l_m) * (k / l_m)) * (k * k))
    else
        tmp = ((l_m / k) * ((l_m / k) / t)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 1.32e+16) {
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (k * k));
	} else {
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 1.32e+16:
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (k * k))
	else:
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 1.32e+16)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) / l_m) * Float64(k / l_m)) * Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l_m / k) * Float64(Float64(l_m / k) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 1.32e+16)
		tmp = 2.0 / ((((k * t) / l_m) * (k / l_m)) * (k * k));
	else
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 1.32e+16], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(N[(l$95$m / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.32e16

   1. Initial program 33.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{{\color{blue}{\ell}}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \color{blue}{\ell}}} \]

      12. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \color{blue}{\ell}\right)} \]

      13. lift-sin.f64 84.0

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 84.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      8. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      17. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      18. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

   7. Applied rewrites 90.6%

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

      2. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{\cos k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2}} \cdot t}{\cos k}} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot \color{blue}{t}}{\cos k}} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   9. Applied rewrites 94.0%

      \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       2. lower-*.f64 80.7

          \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   12. Applied rewrites 80.7%

       \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   if 1.32e16 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      7. lower-ratio-of-squares.f64 48.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
   9. Step-by-step derivation

      1. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      5. lower-/.f64 48.6

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 48.6%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       3. associate-/l* N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       5. lower-/.f64 49.1

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]

   12. Applied rewrites 49.1%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{k \cdot k}\\ \mathbf{if}\;k \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{l\_m}{k} \cdot \frac{\frac{l\_m}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (/ l_m (* k k))))
   (if (<= k 1.32e+16)
     (* (/ 2.0 t) (* t_1 t_1))
     (* (* (/ l_m k) (/ (/ l_m k) t)) -0.3333333333333333))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = l_m / (k * k);
	double tmp;
	if (k <= 1.32e+16) {
		tmp = (2.0 / t) * (t_1 * t_1);
	} else {
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l_m / (k * k)
    if (k <= 1.32d+16) then
        tmp = (2.0d0 / t) * (t_1 * t_1)
    else
        tmp = ((l_m / k) * ((l_m / k) / t)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double t_1 = l_m / (k * k);
	double tmp;
	if (k <= 1.32e+16) {
		tmp = (2.0 / t) * (t_1 * t_1);
	} else {
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = l_m / (k * k)
	tmp = 0
	if k <= 1.32e+16:
		tmp = (2.0 / t) * (t_1 * t_1)
	else:
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(l_m / Float64(k * k))
	tmp = 0.0
	if (k <= 1.32e+16)
		tmp = Float64(Float64(2.0 / t) * Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(Float64(l_m / k) * Float64(Float64(l_m / k) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = l_m / (k * k);
	tmp = 0.0;
	if (k <= 1.32e+16)
		tmp = (2.0 / t) * (t_1 * t_1);
	else
		tmp = ((l_m / k) * ((l_m / k) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(l$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.32e+16], N[(N[(2.0 / t), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(N[(l$95$m / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.32e16

   1. Initial program 33.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{4}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\color{blue}{{\ell}^{2}}}{{k}^{4}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{\color{blue}{k}}^{4}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 45.8

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. pow2 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left({k}^{\color{blue}{2}}\right)\right) \]

      3. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{k}^{2}}\right) \]

      7. pow2 N/A

         \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2}}}\right) \]

      10. pow2 N/A

          \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{k}}\right) \]

      11. lift-*.f64 78.5

          \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{k}}\right) \]

   7. Applied rewrites 78.5%

      \[\leadsto \frac{2}{t} \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]

   if 1.32e16 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      7. lower-ratio-of-squares.f64 48.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
   9. Step-by-step derivation

      1. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      5. lower-/.f64 48.6

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 48.6%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       3. associate-/l* N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       5. lower-/.f64 49.1

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]

   12. Applied rewrites 49.1%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 72.8% accurate, 9.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\mathsf{ratio\_of\_squares}\left(k, l\_m\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{l\_m}{k} \cdot \frac{\frac{l\_m}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1.32e+16)
   (/ 2.0 (* (ratio-of-squares k l_m) (* (* k k) t)))
   (* (* (/ l_m k) (/ (/ l_m k) t)) -0.3333333333333333)))

Derivation

1. Split input into 2 regimes

2. if k < 1.32e16

   1. Initial program 33.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{{\color{blue}{\ell}}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \color{blue}{\ell}}} \]

      12. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \color{blue}{\ell}\right)} \]

      13. lift-sin.f64 84.0

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 84.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      8. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      17. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      18. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

   7. Applied rewrites 90.6%

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left({k}^{2} \cdot t\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. lift-*.f64 78.1

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   10. Applied rewrites 78.1%

       \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

   if 1.32e16 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      7. lower-ratio-of-squares.f64 48.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
   9. Step-by-step derivation

      1. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      2. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

      5. lower-/.f64 48.6

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]

   10. Applied rewrites 48.6%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       3. associate-/l* N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \frac{-1}{3} \]

       5. lower-/.f64 49.1

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]

   12. Applied rewrites 49.1%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 72.7% accurate, 11.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\mathsf{ratio\_of\_squares}\left(k, l\_m\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right)}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1.32e+16)
   (/ 2.0 (* (ratio-of-squares k l_m) (* (* k k) t)))
   (* (/ (ratio-of-squares l_m k) t) -0.3333333333333333)))

Derivation

1. Split input into 2 regimes

2. if k < 1.32e16

   1. Initial program 33.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{{\color{blue}{\ell}}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \color{blue}{\ell}}} \]

      12. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \color{blue}{\ell}\right)} \]

      13. lift-sin.f64 84.0

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   5. Applied rewrites 84.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\sin k}, \ell\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      5. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      8. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      17. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      18. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

   7. Applied rewrites 90.6%

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left({k}^{2} \cdot t\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. lift-*.f64 78.1

         \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   10. Applied rewrites 78.1%

       \[\leadsto \frac{2}{\mathsf{ratio\_of\_squares}\left(k, \ell\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

   if 1.32e16 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      7. lower-ratio-of-squares.f64 48.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, \left(k \cdot k\right)\right)}{t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right)}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1.32e+16)
   (* (/ (ratio-of-squares l_m (* k k)) t) 2.0)
   (* (/ (ratio-of-squares l_m k) t) -0.3333333333333333)))

Derivation

1. Split input into 2 regimes

2. if k < 1.32e16

   1. Initial program 33.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{4}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\color{blue}{{\ell}^{2}}}{{k}^{4}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{\color{blue}{k}}^{4}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 45.8

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

   7. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \cdot 2} \]

   if 1.32e16 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      7. lower-ratio-of-squares.f64 48.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \left(k \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right)}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1.32e+16)
   (* (/ 2.0 t) (ratio-of-squares l_m (* k k)))
   (* (/ (ratio-of-squares l_m k) t) -0.3333333333333333)))

Derivation

1. Split input into 2 regimes

2. if k < 1.32e16

   1. Initial program 33.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{4}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\color{blue}{{\ell}^{2}}}{{k}^{4}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{\color{blue}{k}}^{4}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 45.8

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

   if 1.32e16 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

      7. lower-ratio-of-squares.f64 48.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 31.9% accurate, 19.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right)}{t} \cdot -0.3333333333333333 \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ (ratio-of-squares l_m k) t) -0.3333333333333333))

Derivation

1. Initial program 31.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 46.7%

   \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right) + -0.3333333333333333 \cdot {\left(\ell \cdot k\right)}^{2}}{t}}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{3} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

   7. lower-ratio-of-squares.f64 28.8

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.3333333333333333 \]

8. Applied rewrites 28.8%

   \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot \color{blue}{-0.3333333333333333} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))