Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 94.1%

Time: 8.9s

Alternatives: 17

Speedup: 9.6×

• 39.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;l\_m \leq 10^{+44}:\\ \;\;\;\;\frac{2}{\frac{t\_1}{l\_m} \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{l\_m}{k}\right)}^{2} \cdot \cos k}{t\_1 \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= l_m 1e+44)
     (/ 2.0 (* (/ t_1 l_m) (/ (* (* k t) k) (* (cos k) l_m))))
     (* (/ (* (pow (/ l_m k) 2.0) (cos k)) (* t_1 t)) 2.0))))

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 <= 1e+44) {
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (cos(k) * l_m)));
	} else {
		tmp = ((pow((l_m / k), 2.0) * cos(k)) / (t_1 * t)) * 2.0;
	}
	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 <= 1d+44) then
        tmp = 2.0d0 / ((t_1 / l_m) * (((k * t) * k) / (cos(k) * l_m)))
    else
        tmp = ((((l_m / k) ** 2.0d0) * cos(k)) / (t_1 * t)) * 2.0d0
    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 <= 1e+44) {
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (Math.cos(k) * l_m)));
	} else {
		tmp = ((Math.pow((l_m / k), 2.0) * Math.cos(k)) / (t_1 * t)) * 2.0;
	}
	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 <= 1e+44:
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (math.cos(k) * l_m)))
	else:
		tmp = ((math.pow((l_m / k), 2.0) * math.cos(k)) / (t_1 * t)) * 2.0
	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 <= 1e+44)
		tmp = Float64(2.0 / Float64(Float64(t_1 / l_m) * Float64(Float64(Float64(k * t) * k) / Float64(cos(k) * l_m))));
	else
		tmp = Float64(Float64(Float64((Float64(l_m / k) ^ 2.0) * cos(k)) / Float64(t_1 * t)) * 2.0);
	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 <= 1e+44)
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (cos(k) * l_m)));
	else
		tmp = ((((l_m / k) ^ 2.0) * cos(k)) / (t_1 * t)) * 2.0;
	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, 1e+44], N[(2.0 / N[(N[(t$95$1 / l$95$m), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(l$95$m / k), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.0000000000000001e44

   1. Initial program 34.1%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

   10. Applied rewrites 93.5%

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

   if 1.0000000000000001e44 < l

   1. Initial program 36.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. lift-sin.f64 N/A

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

      19. lift-pow.f64 N/A

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

   8. Applied rewrites 95.0%

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

Alternative 2: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;l\_m \cdot l\_m \leq 10^{+88}:\\ \;\;\;\;\frac{2}{\frac{t\_1}{l\_m} \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{l\_m}{k} \cdot \frac{l\_m}{k}\right) \cdot \frac{\cos k}{t\_1 \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= (* l_m l_m) 1e+88)
     (/ 2.0 (* (/ t_1 l_m) (/ (* (* k t) k) (* (cos k) l_m))))
     (* (* (* (/ l_m k) (/ l_m k)) (/ (cos k) (* t_1 t))) 2.0))))

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 * l_m) <= 1e+88) {
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (cos(k) * l_m)));
	} else {
		tmp = (((l_m / k) * (l_m / k)) * (cos(k) / (t_1 * t))) * 2.0;
	}
	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 * l_m) <= 1d+88) then
        tmp = 2.0d0 / ((t_1 / l_m) * (((k * t) * k) / (cos(k) * l_m)))
    else
        tmp = (((l_m / k) * (l_m / k)) * (cos(k) / (t_1 * t))) * 2.0d0
    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 * l_m) <= 1e+88) {
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (Math.cos(k) * l_m)));
	} else {
		tmp = (((l_m / k) * (l_m / k)) * (Math.cos(k) / (t_1 * t))) * 2.0;
	}
	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 * l_m) <= 1e+88:
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (math.cos(k) * l_m)))
	else:
		tmp = (((l_m / k) * (l_m / k)) * (math.cos(k) / (t_1 * t))) * 2.0
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+88)
		tmp = Float64(2.0 / Float64(Float64(t_1 / l_m) * Float64(Float64(Float64(k * t) * k) / Float64(cos(k) * l_m))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / k) * Float64(l_m / k)) * Float64(cos(k) / Float64(t_1 * t))) * 2.0);
	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 * l_m) <= 1e+88)
		tmp = 2.0 / ((t_1 / l_m) * (((k * t) * k) / (cos(k) * l_m)));
	else
		tmp = (((l_m / k) * (l_m / k)) * (cos(k) / (t_1 * t))) * 2.0;
	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[N[(l$95$m * l$95$m), $MachinePrecision], 1e+88], N[(2.0 / N[(N[(t$95$1 / l$95$m), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999959e87

   1. Initial program 34.1%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

   10. Applied rewrites 93.5%

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

   if 9.99999999999999959e87 < (*.f64 l l)

   1. Initial program 36.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.0

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

   8. Applied rewrites 95.0%

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

Alternative 3: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;l\_m \cdot l\_m \leq 10^{+70}:\\ \;\;\;\;\frac{2}{\frac{t\_1}{l\_m} \cdot \frac{\left(k \cdot k\right) \cdot t}{\cos k \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{l\_m}{k} \cdot \frac{l\_m}{k}\right) \cdot \frac{\cos k}{t\_1 \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= (* l_m l_m) 1e+70)
     (/ 2.0 (* (/ t_1 l_m) (/ (* (* k k) t) (* (cos k) l_m))))
     (* (* (* (/ l_m k) (/ l_m k)) (/ (cos k) (* t_1 t))) 2.0))))

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 * l_m) <= 1e+70) {
		tmp = 2.0 / ((t_1 / l_m) * (((k * k) * t) / (cos(k) * l_m)));
	} else {
		tmp = (((l_m / k) * (l_m / k)) * (cos(k) / (t_1 * t))) * 2.0;
	}
	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 * l_m) <= 1d+70) then
        tmp = 2.0d0 / ((t_1 / l_m) * (((k * k) * t) / (cos(k) * l_m)))
    else
        tmp = (((l_m / k) * (l_m / k)) * (cos(k) / (t_1 * t))) * 2.0d0
    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 * l_m) <= 1e+70) {
		tmp = 2.0 / ((t_1 / l_m) * (((k * k) * t) / (Math.cos(k) * l_m)));
	} else {
		tmp = (((l_m / k) * (l_m / k)) * (Math.cos(k) / (t_1 * t))) * 2.0;
	}
	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 * l_m) <= 1e+70:
		tmp = 2.0 / ((t_1 / l_m) * (((k * k) * t) / (math.cos(k) * l_m)))
	else:
		tmp = (((l_m / k) * (l_m / k)) * (math.cos(k) / (t_1 * t))) * 2.0
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+70)
		tmp = Float64(2.0 / Float64(Float64(t_1 / l_m) * Float64(Float64(Float64(k * k) * t) / Float64(cos(k) * l_m))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / k) * Float64(l_m / k)) * Float64(cos(k) / Float64(t_1 * t))) * 2.0);
	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 * l_m) <= 1e+70)
		tmp = 2.0 / ((t_1 / l_m) * (((k * k) * t) / (cos(k) * l_m)));
	else
		tmp = (((l_m / k) * (l_m / k)) * (cos(k) / (t_1 * t))) * 2.0;
	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[N[(l$95$m * l$95$m), $MachinePrecision], 1e+70], N[(2.0 / N[(N[(t$95$1 / l$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000007e70

   1. Initial program 34.1%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 91.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 92.2%

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

   if 1.00000000000000007e70 < (*.f64 l l)

   1. Initial program 36.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 70.0

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

   4. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.0

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

   8. Applied rewrites 95.0%

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

Alternative 4: 85.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-173}:\\ \;\;\;\;\frac{2}{\frac{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{l\_m} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{l\_m \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{l\_m}{k} \cdot \frac{l\_m}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 3.3e-173)
   (/
    2.0
    (/
     (* (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) l_m) (* k (* k t)))
     (* l_m (cos k))))
   (* (* (* (/ l_m k) (/ l_m k)) (/ (cos k) (* (pow (sin k) 2.0) t))) 2.0)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 3.3e-173) {
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * cos(k)));
	} else {
		tmp = (((l_m / k) * (l_m / k)) * (cos(k) / (pow(sin(k), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.3d-173) then
        tmp = 2.0d0 / ((((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) / l_m) * (k * (k * t))) / (l_m * cos(k)))
    else
        tmp = (((l_m / k) * (l_m / k)) * (cos(k) / ((sin(k) ** 2.0d0) * t))) * 2.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 3.3e-173) {
		tmp = 2.0 / ((((0.5 - (0.5 * Math.cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * Math.cos(k)));
	} else {
		tmp = (((l_m / k) * (l_m / k)) * (Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if t <= 3.3e-173:
		tmp = 2.0 / ((((0.5 - (0.5 * math.cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * math.cos(k)))
	else:
		tmp = (((l_m / k) * (l_m / k)) * (math.cos(k) / (math.pow(math.sin(k), 2.0) * t))) * 2.0
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 3.3e-173)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / l_m) * Float64(k * Float64(k * t))) / Float64(l_m * cos(k))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / k) * Float64(l_m / k)) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * t))) * 2.0);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (t <= 3.3e-173)
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * cos(k)));
	else
		tmp = (((l_m / k) * (l_m / k)) * (cos(k) / ((sin(k) ^ 2.0) * t))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 3.3e-173], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.3000000000000003e-173

   1. Initial program 33.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.2

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

   4. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 81.0

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

   10. Applied rewrites 81.0%

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

   if 3.3000000000000003e-173 < t

   1. Initial program 38.7%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.6

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

   8. Applied rewrites 93.6%

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

Alternative 5: 83.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{2}{\frac{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{l\_m} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{l\_m \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(l\_m \cdot \frac{l\_m}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 2.7e-145)
   (/
    2.0
    (/
     (* (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) l_m) (* k (* k t)))
     (* l_m (cos k))))
   (* (* (* l_m (/ l_m (* k k))) (/ (cos k) (* (pow (sin k) 2.0) t))) 2.0)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 2.7e-145) {
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * cos(k)));
	} else {
		tmp = ((l_m * (l_m / (k * k))) * (cos(k) / (pow(sin(k), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.7d-145) then
        tmp = 2.0d0 / ((((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) / l_m) * (k * (k * t))) / (l_m * cos(k)))
    else
        tmp = ((l_m * (l_m / (k * k))) * (cos(k) / ((sin(k) ** 2.0d0) * t))) * 2.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 2.7e-145) {
		tmp = 2.0 / ((((0.5 - (0.5 * Math.cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * Math.cos(k)));
	} else {
		tmp = ((l_m * (l_m / (k * k))) * (Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if t <= 2.7e-145:
		tmp = 2.0 / ((((0.5 - (0.5 * math.cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * math.cos(k)))
	else:
		tmp = ((l_m * (l_m / (k * k))) * (math.cos(k) / (math.pow(math.sin(k), 2.0) * t))) * 2.0
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 2.7e-145)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / l_m) * Float64(k * Float64(k * t))) / Float64(l_m * cos(k))));
	else
		tmp = Float64(Float64(Float64(l_m * Float64(l_m / Float64(k * k))) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * t))) * 2.0);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (t <= 2.7e-145)
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) / l_m) * (k * (k * t))) / (l_m * cos(k)));
	else
		tmp = ((l_m * (l_m / (k * k))) * (cos(k) / ((sin(k) ^ 2.0) * t))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 2.7e-145], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * N[(l$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.7e-145

   1. Initial program 33.3%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.3

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

   4. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 81.2

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

   10. Applied rewrites 81.2%

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

   if 2.7e-145 < t

   1. Initial program 38.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 87.0

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

   8. Applied rewrites 87.0%

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

Alternative 6: 78.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{l\_m} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{l\_m}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{l\_m \cdot l\_m}{k \cdot k} \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{l\_m}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 9e-9)
   (/ 2.0 (/ (* (/ (pow (sin k) 2.0) l_m) (* k (* k t))) l_m))
   (if (<= k 2e+134)
     (*
      (*
       (/ (* l_m l_m) (* k k))
       (/ (cos k) (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)))
      2.0)
     (* (pow (/ l_m k) 2.0) (/ -0.3333333333333333 t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 9e-9) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / l_m) * (k * (k * t))) / l_m);
	} else if (k <= 2e+134) {
		tmp = (((l_m * l_m) / (k * k)) * (cos(k) / ((0.5 - (0.5 * cos((2.0 * k)))) * t))) * 2.0;
	} else {
		tmp = pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d-9) then
        tmp = 2.0d0 / ((((sin(k) ** 2.0d0) / l_m) * (k * (k * t))) / l_m)
    else if (k <= 2d+134) then
        tmp = (((l_m * l_m) / (k * k)) * (cos(k) / ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * t))) * 2.0d0
    else
        tmp = ((l_m / k) ** 2.0d0) * ((-0.3333333333333333d0) / t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 9e-9) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / l_m) * (k * (k * t))) / l_m);
	} else if (k <= 2e+134) {
		tmp = (((l_m * l_m) / (k * k)) * (Math.cos(k) / ((0.5 - (0.5 * Math.cos((2.0 * k)))) * t))) * 2.0;
	} else {
		tmp = Math.pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 9e-9:
		tmp = 2.0 / (((math.pow(math.sin(k), 2.0) / l_m) * (k * (k * t))) / l_m)
	elif k <= 2e+134:
		tmp = (((l_m * l_m) / (k * k)) * (math.cos(k) / ((0.5 - (0.5 * math.cos((2.0 * k)))) * t))) * 2.0
	else:
		tmp = math.pow((l_m / k), 2.0) * (-0.3333333333333333 / t)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 9e-9)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / l_m) * Float64(k * Float64(k * t))) / l_m));
	elseif (k <= 2e+134)
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) / Float64(k * k)) * Float64(cos(k) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t))) * 2.0);
	else
		tmp = Float64((Float64(l_m / k) ^ 2.0) * Float64(-0.3333333333333333 / t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 9e-9)
		tmp = 2.0 / ((((sin(k) ^ 2.0) / l_m) * (k * (k * t))) / l_m);
	elseif (k <= 2e+134)
		tmp = (((l_m * l_m) / (k * k)) * (cos(k) / ((0.5 - (0.5 * cos((2.0 * k)))) * t))) * 2.0;
	else
		tmp = ((l_m / k) ^ 2.0) * (-0.3333333333333333 / t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 9e-9], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e+134], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Power[N[(l$95$m / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.99999999999999953e-9

   1. Initial program 36.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.6

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 80.0%

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

   if 8.99999999999999953e-9 < k < 1.99999999999999984e134

   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. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 81.2

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 81.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 80.5

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

   8. Applied rewrites 80.5%

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

   if 1.99999999999999984e134 < k

   1. Initial program 37.9%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-pow.f64 32.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   5. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 60.3

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 60.3%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 60.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   9. Applied rewrites 60.3%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*l/ N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. pow-to-exp N/A

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

       12. lift-log.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-exp.f64 N/A

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

   11. Applied rewrites 65.6%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{l\_m} \cdot t\_1}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{l\_m} \cdot t\_1}{l\_m \cdot \cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= k 9e-9)
     (/ 2.0 (/ (* (/ (pow (sin k) 2.0) l_m) t_1) l_m))
     (/
      2.0
      (/ (* (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) l_m) t_1) (* l_m (cos k)))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 9e-9) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / l_m) * t_1) / l_m);
	} else {
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) / l_m) * t_1) / (l_m * 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 * (k * t)
    if (k <= 9d-9) then
        tmp = 2.0d0 / ((((sin(k) ** 2.0d0) / l_m) * t_1) / l_m)
    else
        tmp = 2.0d0 / ((((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) / l_m) * t_1) / (l_m * 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 * (k * t);
	double tmp;
	if (k <= 9e-9) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / l_m) * t_1) / l_m);
	} else {
		tmp = 2.0 / ((((0.5 - (0.5 * Math.cos((2.0 * k)))) / l_m) * t_1) / (l_m * Math.cos(k)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	t_1 = k * (k * t)
	tmp = 0
	if k <= 9e-9:
		tmp = 2.0 / (((math.pow(math.sin(k), 2.0) / l_m) * t_1) / l_m)
	else:
		tmp = 2.0 / ((((0.5 - (0.5 * math.cos((2.0 * k)))) / l_m) * t_1) / (l_m * math.cos(k)))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 9e-9)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / l_m) * t_1) / l_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / l_m) * t_1) / Float64(l_m * cos(k))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (k <= 9e-9)
		tmp = 2.0 / ((((sin(k) ^ 2.0) / l_m) * t_1) / l_m);
	else
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) / l_m) * t_1) / (l_m * 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[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 9e-9], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(l$95$m * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999953e-9

   1. Initial program 36.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.6

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 80.0%

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

   if 8.99999999999999953e-9 < k

   1. Initial program 30.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 71.1

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

   4. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 83.6

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

   10. Applied rewrites 83.6%

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

Alternative 8: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 2120000000:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{l\_m} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{l\_m \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{l\_m \cdot l\_m}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 2120000000.0)
   (/ 2.0 (/ (* (/ (* k k) l_m) (* (* k k) t)) (* l_m (cos k))))
   (* (/ 2.0 (* k (* k t))) (/ (* l_m l_m) (pow (sin k) 2.0)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 2120000000.0) {
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * cos(k)));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l_m * l_m) / pow(sin(k), 2.0));
	}
	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 <= 2120000000.0d0) then
        tmp = 2.0d0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * cos(k)))
    else
        tmp = (2.0d0 / (k * (k * t))) * ((l_m * l_m) / (sin(k) ** 2.0d0))
    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 <= 2120000000.0) {
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * Math.cos(k)));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l_m * l_m) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 2120000000.0:
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * math.cos(k)))
	else:
		tmp = (2.0 / (k * (k * t))) * ((l_m * l_m) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 2120000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l_m) * Float64(Float64(k * k) * t)) / Float64(l_m * cos(k))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l_m * l_m) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 2120000000.0)
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * cos(k)));
	else
		tmp = (2.0 / (k * (k * t))) * ((l_m * l_m) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 2120000000.0], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.12e9

   1. Initial program 36.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 76.7

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 79.4

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

   9. Applied rewrites 79.4%

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

   if 2.12e9 < k

   1. Initial program 31.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 54.1

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

   7. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 55.7

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

   9. Applied rewrites 55.7%

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

Alternative 9: 74.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = 2.0d0 / ((((sin(k) ** 2.0d0) / l_m) * (k * (k * t))) / l_m)
end function

Java

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

Python

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

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / l_m) * Float64(k * Float64(k * t))) / l_m))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = 2.0 / ((((sin(k) ^ 2.0) / l_m) * (k * (k * t))) / l_m);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 75.1

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

4. Applied rewrites 75.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\ell \cdot \ell}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

   5. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

   9. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k}}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\cos k}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot t}{\cos k}} \]

   13. frac-times N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left({k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left({k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

6. Applied rewrites 84.5%

   \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell \cdot \cos k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell \cdot \cos k}} \]

   5. lower-*.f64 88.2

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell \cdot \cos k}} \]

8. Applied rewrites 88.2%

   \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell \cdot \cos k}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell}} \]
10. Step-by-step derivation

11. Applied rewrites 74.2%

    \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell}} \]
12. Add Preprocessing

Alternative 10: 61.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 2120000000:\\ \;\;\;\;l\_m \cdot \left(l\_m \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}\right)\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{l\_m \cdot l\_m}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{l\_m}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 2120000000.0)
   (* l_m (* l_m (/ (fma (* k k) -0.3333333333333333 2.0) (* (pow k 4.0) t))))
   (if (<= k 4.5e+132)
     (* (/ 2.0 (* (* k k) t)) (/ (* l_m l_m) (- 0.5 (* 0.5 (cos (* 2.0 k))))))
     (* (pow (/ l_m k) 2.0) (/ -0.3333333333333333 t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 2120000000.0) {
		tmp = l_m * (l_m * (fma((k * k), -0.3333333333333333, 2.0) / (pow(k, 4.0) * t)));
	} else if (k <= 4.5e+132) {
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / (0.5 - (0.5 * cos((2.0 * k)))));
	} else {
		tmp = pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 2120000000.0)
		tmp = Float64(l_m * Float64(l_m * Float64(fma(Float64(k * k), -0.3333333333333333, 2.0) / Float64((k ^ 4.0) * t))));
	elseif (k <= 4.5e+132)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l_m * l_m) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))));
	else
		tmp = Float64((Float64(l_m / k) ^ 2.0) * Float64(-0.3333333333333333 / t));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 2120000000.0], N[(l$95$m * N[(l$95$m * N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.5e+132], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l$95$m / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.12e9

   1. Initial program 36.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 55.2

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\color{blue}{{k}^{4}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{\color{blue}{k}}^{4}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{\color{blue}{4}}} \]

      9. associate-/l/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right) + \frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2}}{\color{blue}{t} \cdot {k}^{4}} \]

      11. pow2 N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2}}{t \cdot {k}^{4}} \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({\ell}^{2} \cdot {k}^{2}\right)}{t \cdot {k}^{4}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t \cdot {k}^{4}} \]

      14. associate-*r* N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}{t \cdot {k}^{4}} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{\color{blue}{t} \cdot {k}^{4}} \]

      16. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   6. Applied rewrites 56.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}}{{k}^{4} \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{\color{blue}{{k}^{4} \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{\color{blue}{k}}^{4} \cdot t} \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{\color{blue}{{k}^{4}} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot \color{blue}{t}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t} \]

      8. associate-*l* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}}\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{\color{blue}{{k}^{4}} \cdot t}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{\color{blue}{k}}^{4} \cdot t}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot t}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot \color{blue}{t}}\right) \]

      15. lift-/.f64 63.3

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\color{blue}{{k}^{4} \cdot t}}\right) \]

   8. Applied rewrites 63.3%

      \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}\right)} \]

   if 2.12e9 < k < 4.49999999999999972e132

   1. Initial program 22.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 80.4

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   4. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}} \]

      2. lift-*.f64 45.5

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}} \]

   7. Applied rewrites 45.5%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\color{blue}{\sin k}}^{2}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{\color{blue}{2}}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \color{blue}{\sin k}} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot k\right)}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)} \]

      8. lower-*.f64 45.5

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \]

   9. Applied rewrites 45.5%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot k\right)}} \]

   if 4.49999999999999972e132 < k

   1. Initial program 37.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 32.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 60.2

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 60.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   9. Applied rewrites 60.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. associate-*l/ N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

       6. pow2 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

       11. pow-to-exp N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       12. lift-log.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       14. lift-exp.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

   11. Applied rewrites 65.5%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-162}:\\ \;\;\;\;l\_m \cdot \left(l\_m \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}\right)\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{l\_m \cdot l\_m}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot t\right) \cdot k}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{l\_m}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 2.1e-162)
   (* l_m (* l_m (/ (fma (* k k) -0.3333333333333333 2.0) (* (pow k 4.0) t))))
   (if (<= k 4.5e+132)
     (* (* (/ (* l_m l_m) (* k k)) (/ (cos k) (* (* k t) k))) 2.0)
     (* (pow (/ l_m k) 2.0) (/ -0.3333333333333333 t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-162) {
		tmp = l_m * (l_m * (fma((k * k), -0.3333333333333333, 2.0) / (pow(k, 4.0) * t)));
	} else if (k <= 4.5e+132) {
		tmp = (((l_m * l_m) / (k * k)) * (cos(k) / ((k * t) * k))) * 2.0;
	} else {
		tmp = pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-162)
		tmp = Float64(l_m * Float64(l_m * Float64(fma(Float64(k * k), -0.3333333333333333, 2.0) / Float64((k ^ 4.0) * t))));
	elseif (k <= 4.5e+132)
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) / Float64(k * k)) * Float64(cos(k) / Float64(Float64(k * t) * k))) * 2.0);
	else
		tmp = Float64((Float64(l_m / k) ^ 2.0) * Float64(-0.3333333333333333 / t));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 2.1e-162], N[(l$95$m * N[(l$95$m * N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.5e+132], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Power[N[(l$95$m / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.1e-162

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 54.1

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\color{blue}{{k}^{4}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{\color{blue}{k}}^{4}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{\color{blue}{4}}} \]

      9. associate-/l/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right) + \frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2}}{\color{blue}{t} \cdot {k}^{4}} \]

      11. pow2 N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2}}{t \cdot {k}^{4}} \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({\ell}^{2} \cdot {k}^{2}\right)}{t \cdot {k}^{4}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t \cdot {k}^{4}} \]

      14. associate-*r* N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}{t \cdot {k}^{4}} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{\color{blue}{t} \cdot {k}^{4}} \]

      16. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   6. Applied rewrites 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}}{{k}^{4} \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{\color{blue}{{k}^{4} \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{\color{blue}{k}}^{4} \cdot t} \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{\color{blue}{{k}^{4}} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot \color{blue}{t}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t} \]

      8. associate-*l* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}}\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{\color{blue}{{k}^{4}} \cdot t}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{\color{blue}{k}}^{4} \cdot t}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot t}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot \color{blue}{t}}\right) \]

      15. lift-/.f64 60.9

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\color{blue}{{k}^{4} \cdot t}}\right) \]

   8. Applied rewrites 60.9%

      \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}\right)} \]

   if 2.1e-162 < k < 4.49999999999999972e132

   1. Initial program 26.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 80.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      9. frac-times N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\color{blue}{\sin k}}^{2}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

   6. Applied rewrites 80.6%

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot \color{blue}{2} \]

   7. Taylor expanded in k around 0

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{k}^{2} \cdot t}\right) \cdot 2 \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right) \cdot 2 \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}\right) \cdot 2 \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot t\right) \cdot k}\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot t\right) \cdot k}\right) \cdot 2 \]

      5. lift-*.f64 65.4

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot t\right) \cdot k}\right) \cdot 2 \]

   9. Applied rewrites 65.4%

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot t\right) \cdot k}\right) \cdot 2 \]

   if 4.49999999999999972e132 < k

   1. Initial program 37.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 32.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 60.2

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 60.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   9. Applied rewrites 60.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. associate-*l/ N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

       6. pow2 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

       11. pow-to-exp N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       12. lift-log.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       14. lift-exp.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

   11. Applied rewrites 65.5%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{l\_m} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{l\_m \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{l\_m}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 6.8e+95)
   (/ 2.0 (/ (* (/ (* k k) l_m) (* (* k k) t)) (* l_m (cos k))))
   (* (pow (/ l_m k) 2.0) (/ -0.3333333333333333 t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 6.8e+95) {
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * cos(k)));
	} else {
		tmp = pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.8d+95) then
        tmp = 2.0d0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * cos(k)))
    else
        tmp = ((l_m / k) ** 2.0d0) * ((-0.3333333333333333d0) / t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 6.8e+95) {
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * Math.cos(k)));
	} else {
		tmp = Math.pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 6.8e+95:
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * math.cos(k)))
	else:
		tmp = math.pow((l_m / k), 2.0) * (-0.3333333333333333 / t)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 6.8e+95)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l_m) * Float64(Float64(k * k) * t)) / Float64(l_m * cos(k))));
	else
		tmp = Float64((Float64(l_m / k) ^ 2.0) * Float64(-0.3333333333333333 / t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 6.8e+95)
		tmp = 2.0 / ((((k * k) / l_m) * ((k * k) * t)) / (l_m * cos(k)));
	else
		tmp = ((l_m / k) ^ 2.0) * (-0.3333333333333333 / t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 6.8e+95], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l$95$m / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000043e95

   1. Initial program 35.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 77.0

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   4. Applied rewrites 77.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\cos k}} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot t}{\cos k}} \]

      13. frac-times N/A

          \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left({k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left({k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

   6. Applied rewrites 87.6%

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

      2. lift-*.f64 76.3

         \[\leadsto \frac{2}{\frac{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   9. Applied rewrites 76.3%

      \[\leadsto \frac{2}{\frac{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   if 6.80000000000000043e95 < k

   1. Initial program 35.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 33.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 58.4

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 58.4%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 58.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. associate-*l/ N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

       6. pow2 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

       11. pow-to-exp N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       12. lift-log.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       14. lift-exp.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

   11. Applied rewrites 63.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;l\_m \cdot \left(l\_m \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}\right)\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.08611111111111111, 0.16666666666666666\right) \cdot t, k \cdot k, t\right)}{l\_m \cdot l\_m} \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{l\_m}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1.95e-146)
   (* l_m (* l_m (/ (fma (* k k) -0.3333333333333333 2.0) (* (pow k 4.0) t))))
   (if (<= k 6.8e+95)
     (/
      2.0
      (*
       (*
        (/
         (fma
          (* (fma (* k k) 0.08611111111111111 0.16666666666666666) t)
          (* k k)
          t)
         (* l_m l_m))
        (* k k))
       (* k k)))
     (* (pow (/ l_m k) 2.0) (/ -0.3333333333333333 t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 1.95e-146) {
		tmp = l_m * (l_m * (fma((k * k), -0.3333333333333333, 2.0) / (pow(k, 4.0) * t)));
	} else if (k <= 6.8e+95) {
		tmp = 2.0 / (((fma((fma((k * k), 0.08611111111111111, 0.16666666666666666) * t), (k * k), t) / (l_m * l_m)) * (k * k)) * (k * k));
	} else {
		tmp = pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 1.95e-146)
		tmp = Float64(l_m * Float64(l_m * Float64(fma(Float64(k * k), -0.3333333333333333, 2.0) / Float64((k ^ 4.0) * t))));
	elseif (k <= 6.8e+95)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(Float64(k * k), 0.08611111111111111, 0.16666666666666666) * t), Float64(k * k), t) / Float64(l_m * l_m)) * Float64(k * k)) * Float64(k * k)));
	else
		tmp = Float64((Float64(l_m / k) ^ 2.0) * Float64(-0.3333333333333333 / t));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 1.95e-146], N[(l$95$m * N[(l$95$m * N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.8e+95], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 0.08611111111111111 + 0.16666666666666666), $MachinePrecision] * t), $MachinePrecision] * N[(k * k), $MachinePrecision] + t), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l$95$m / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.95000000000000001e-146

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 54.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\color{blue}{{k}^{4}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{\color{blue}{k}}^{4}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{4}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{t}}{{k}^{\color{blue}{4}}} \]

      9. associate-/l/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + 2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right) + \frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2}}{\color{blue}{t} \cdot {k}^{4}} \]

      11. pow2 N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2}}{t \cdot {k}^{4}} \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({\ell}^{2} \cdot {k}^{2}\right)}{t \cdot {k}^{4}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t \cdot {k}^{4}} \]

      14. associate-*r* N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}{t \cdot {k}^{4}} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{\color{blue}{t} \cdot {k}^{4}} \]

      16. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   6. Applied rewrites 53.7%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}}{{k}^{4} \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot t}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{\color{blue}{{k}^{4} \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{\color{blue}{k}}^{4} \cdot t} \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{\color{blue}{{k}^{4}} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot \color{blue}{t}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t} \]

      8. associate-*l* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot \frac{-1}{3} + 2}{{k}^{4} \cdot t}}\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{\color{blue}{{k}^{4}} \cdot t}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{\color{blue}{k}}^{4} \cdot t}\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot t}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 2\right)}{{k}^{4} \cdot \color{blue}{t}}\right) \]

      15. lift-/.f64 61.1

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\color{blue}{{k}^{4} \cdot t}}\right) \]

   8. Applied rewrites 61.1%

      \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{{k}^{4} \cdot t}\right)} \]

   if 1.95000000000000001e-146 < k < 6.80000000000000043e95

   1. Initial program 26.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right) + \frac{t}{{\ell}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right) + \frac{t}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{4}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right) + \frac{t}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{4}}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.16666666666666666, t, 0.08611111111111111 \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot {k}^{4}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{6}, t, \frac{31}{360} \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot {k}^{\color{blue}{4}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{6}, t, \frac{31}{360} \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot {k}^{\left(2 + \color{blue}{2}\right)}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{6}, t, \frac{31}{360} \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{{k}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{6}, t, \frac{31}{360} \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{{k}^{2}}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{6}, t, \frac{31}{360} \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot \left(\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{6}, t, \frac{31}{360} \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot \left(\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{6}, t, \frac{31}{360} \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

      8. lift-*.f64 59.6

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.16666666666666666, t, 0.08611111111111111 \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

   6. Applied rewrites 59.6%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.16666666666666666, t, 0.08611111111111111 \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{t}{\ell \cdot \ell}\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 62.3%

      \[\leadsto \frac{2}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.08611111111111111, 0.16666666666666666\right) \cdot t, k \cdot k, t\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   if 6.80000000000000043e95 < k

   1. Initial program 35.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 33.9

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 33.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 58.4

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 58.4%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 58.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. associate-*l/ N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

       6. pow2 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

       11. pow-to-exp N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       12. lift-log.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       14. lift-exp.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

   11. Applied rewrites 63.0%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{l\_m \cdot l\_m}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{l\_m}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.5e+132)
   (* (/ 2.0 (* (* k k) t)) (/ (* l_m l_m) (* k k)))
   (* (pow (/ l_m k) 2.0) (/ -0.3333333333333333 t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.5e+132) {
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k));
	} else {
		tmp = pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.5d+132) then
        tmp = (2.0d0 / ((k * k) * t)) * ((l_m * l_m) / (k * k))
    else
        tmp = ((l_m / k) ** 2.0d0) * ((-0.3333333333333333d0) / t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.5e+132) {
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k));
	} else {
		tmp = Math.pow((l_m / k), 2.0) * (-0.3333333333333333 / t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if k <= 4.5e+132:
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k))
	else:
		tmp = math.pow((l_m / k), 2.0) * (-0.3333333333333333 / t)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.5e+132)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l_m * l_m) / Float64(k * k)));
	else
		tmp = Float64((Float64(l_m / k) ^ 2.0) * Float64(-0.3333333333333333 / t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (k <= 4.5e+132)
		tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k));
	else
		tmp = ((l_m / k) ^ 2.0) * (-0.3333333333333333 / t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.5e+132], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l$95$m / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.3333333333333333 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.49999999999999972e132

   1. Initial program 34.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 77.1

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   4. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{k}^{\color{blue}{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      4. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

      5. lift-*.f64 66.0

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   7. Applied rewrites 66.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

   if 4.49999999999999972e132 < k

   1. Initial program 37.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

      14. lower-pow.f64 32.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 60.2

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      13. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

      15. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

      17. lift-*.f64 60.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

   9. Applied rewrites 60.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       4. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

       5. associate-*l/ N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

       6. pow2 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

       11. pow-to-exp N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       12. lift-log.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

       14. lift-exp.f64 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

   11. Applied rewrites 65.5%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{l\_m \cdot l\_m}{k \cdot k} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ 2.0 (* (* k k) t)) (/ (* l_m l_m) (* k k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k));
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (2.0d0 / ((k * k) * t)) * ((l_m * l_m) / (k * k))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l_m * l_m) / Float64(k * k)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (2.0 / ((k * k) * t)) * ((l_m * l_m) / (k * k));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 75.1

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

4. Applied rewrites 75.1%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

5. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{k}^{\color{blue}{2}}} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   4. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   5. lift-*.f64 65.1

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

7. Applied rewrites 65.1%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]
8. Add Preprocessing

Alternative 16: 31.4% accurate, 14.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \left(l\_m \cdot \frac{l\_m}{\left(k \cdot t\right) \cdot k}\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (* l_m (/ l_m (* (* k t) k))) -0.3333333333333333))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (l_m * (l_m / ((k * t) * k))) * -0.3333333333333333;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (l_m * (l_m / ((k * t) * k))) * (-0.3333333333333333d0)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (l_m * (l_m / ((k * t) * k))) * -0.3333333333333333;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (l_m * (l_m / ((k * t) * k))) * -0.3333333333333333

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(l_m * Float64(l_m / Float64(Float64(k * t) * k))) * -0.3333333333333333)
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (l_m * (l_m / ((k * t) * k))) * -0.3333333333333333;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(l$95$m * N[(l$95$m / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   4. div-add-rev N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   8. pow-prod-down N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   14. lower-pow.f64 50.2

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

4. Applied rewrites 50.2%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   9. lift-*.f64 30.0

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 30.0%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   17. lift-*.f64 30.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

9. Applied rewrites 30.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    3. associate-/l* N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    6. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    7. pow2 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{-1}{3} \]

    8. lower-/.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{-1}{3} \]

    9. pow2 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{-1}{3} \]

    10. associate-*r* N/A

        \[\leadsto \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot \frac{-1}{3} \]

    11. *-commutative N/A

        \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot k}\right) \cdot \frac{-1}{3} \]

    12. lower-*.f64 N/A

        \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot k}\right) \cdot \frac{-1}{3} \]

    13. lift-*.f64 31.4

        \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot k}\right) \cdot -0.3333333333333333 \]

11. Applied rewrites 31.4%

    \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot k}\right) \cdot -0.3333333333333333 \]
12. Add Preprocessing

Alternative 17: 30.4% accurate, 14.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \left(l\_m \cdot l\_m\right) \cdot \frac{-0.3333333333333333}{\left(k \cdot t\right) \cdot k} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (* l_m l_m) (/ -0.3333333333333333 (* (* k t) k))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (l_m * l_m) * (-0.3333333333333333 / ((k * t) * k));
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (l_m * l_m) * ((-0.3333333333333333d0) / ((k * t) * k))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (l_m * l_m) * (-0.3333333333333333 / ((k * t) * k));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (l_m * l_m) * (-0.3333333333333333 / ((k * t) * k))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(l_m * l_m) * Float64(-0.3333333333333333 / Float64(Float64(k * t) * k)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (l_m * l_m) * (-0.3333333333333333 / ((k * t) * k));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t} + \frac{2 \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   4. div-add-rev N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {k}^{2} \cdot {\ell}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\ell}^{2} \cdot {k}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   8. pow-prod-down N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot {\ell}^{2}\right)}{t}}{{k}^{4}} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}} \]

   14. lower-pow.f64 50.2

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{\color{blue}{4}}} \]

4. Applied rewrites 50.2%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, {\left(\ell \cdot k\right)}^{2}, 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{{k}^{4}}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   9. lift-*.f64 30.0

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 30.0%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   17. lift-*.f64 30.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

9. Applied rewrites 30.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    3. lift-/.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    4. pow2 N/A

       \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    5. associate-*l/ N/A

       \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

    8. pow2 N/A

       \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

    9. pow-to-exp N/A

       \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{e^{\log k \cdot 2} \cdot t} \]

    10. associate-/l* N/A

        \[\leadsto {\ell}^{2} \cdot \frac{\frac{-1}{3}}{\color{blue}{e^{\log k \cdot 2} \cdot t}} \]

    11. lower-*.f64 N/A

        \[\leadsto {\ell}^{2} \cdot \frac{\frac{-1}{3}}{\color{blue}{e^{\log k \cdot 2} \cdot t}} \]

    12. pow2 N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\color{blue}{e^{\log k \cdot 2}} \cdot t} \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\color{blue}{e^{\log k \cdot 2}} \cdot t} \]

    14. pow-to-exp N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{{k}^{2} \cdot t} \]

    15. lower-/.f64 N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{{k}^{2} \cdot \color{blue}{t}} \]

    16. pow2 N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

    17. associate-*r* N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

    18. *-commutative N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\left(k \cdot t\right) \cdot k} \]

    19. lower-*.f64 N/A

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\left(k \cdot t\right) \cdot k} \]

    20. lift-*.f64 30.4

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\left(k \cdot t\right) \cdot k} \]

11. Applied rewrites 30.4%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot t\right) \cdot k}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025097 
(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))))