Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 92.1%

Time: 7.9s

Alternatives: 18

Speedup: 8.6×

• 40.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(\frac{t}{\cos k\_m} \cdot \frac{\frac{t\_1}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_1 \cdot \left(k\_m \cdot \frac{k\_m \cdot t}{\ell}\right)}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= k_m 1.7e-99)
     (/ 2.0 (* (* k_m k_m) (* (/ t (cos k_m)) (/ (/ t_1 l) l))))
     (/ 2.0 (/ (* t_1 (* k_m (/ (* k_m t) l))) (* (cos k_m) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 1.7e-99) {
		tmp = 2.0 / ((k_m * k_m) * ((t / cos(k_m)) * ((t_1 / l) / l)));
	} else {
		tmp = 2.0 / ((t_1 * (k_m * ((k_m * t) / l))) / (cos(k_m) * l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    if (k_m <= 1.7d-99) then
        tmp = 2.0d0 / ((k_m * k_m) * ((t / cos(k_m)) * ((t_1 / l) / l)))
    else
        tmp = 2.0d0 / ((t_1 * (k_m * ((k_m * t) / l))) / (cos(k_m) * l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 1.7e-99) {
		tmp = 2.0 / ((k_m * k_m) * ((t / Math.cos(k_m)) * ((t_1 / l) / l)));
	} else {
		tmp = 2.0 / ((t_1 * (k_m * ((k_m * t) / l))) / (Math.cos(k_m) * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if k_m <= 1.7e-99:
		tmp = 2.0 / ((k_m * k_m) * ((t / math.cos(k_m)) * ((t_1 / l) / l)))
	else:
		tmp = 2.0 / ((t_1 * (k_m * ((k_m * t) / l))) / (math.cos(k_m) * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 1.7e-99)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t / cos(k_m)) * Float64(Float64(t_1 / l) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(k_m * Float64(Float64(k_m * t) / l))) / Float64(cos(k_m) * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 1.7e-99)
		tmp = 2.0 / ((k_m * k_m) * ((t / cos(k_m)) * ((t_1 / l) / l)));
	else
		tmp = 2.0 / ((t_1 * (k_m * ((k_m * t) / l))) / (cos(k_m) * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 1.7e-99], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.70000000000000003e-99

   1. Initial program 47.2%

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

   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 74.4

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

   4. Applied rewrites 74.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 92.1%

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

   if 1.70000000000000003e-99 < k

   1. Initial program 30.4%

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

   2. 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 74.0

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

   4. Applied rewrites 74.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 92.0%

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

Alternative 2: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot t\right)}{\mathsf{fma}\left(-0.5, k\_m \cdot k\_m, 1\right)} \cdot \left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k\_m}^{2} \cdot \left(k\_m \cdot \frac{k\_m \cdot t}{\ell}\right)}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-101)
   (/
    2.0
    (*
     (/ (* k_m (* k_m t)) (fma -0.5 (* k_m k_m) 1.0))
     (* (/ k_m l) (/ k_m l))))
   (/
    2.0
    (/ (* (pow (sin k_m) 2.0) (* k_m (/ (* k_m t) l))) (* (cos k_m) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-101) {
		tmp = 2.0 / (((k_m * (k_m * t)) / fma(-0.5, (k_m * k_m), 1.0)) * ((k_m / l) * (k_m / l)));
	} else {
		tmp = 2.0 / ((pow(sin(k_m), 2.0) * (k_m * ((k_m * t) / l))) / (cos(k_m) * l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-101)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m * t)) / fma(-0.5, Float64(k_m * k_m), 1.0)) * Float64(Float64(k_m / l) * Float64(k_m / l))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k_m) ^ 2.0) * Float64(k_m * Float64(Float64(k_m * t) / l))) / Float64(cos(k_m) * l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-101], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-101

   1. Initial program 47.2%

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

   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 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 92.0

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   10. Applied rewrites 92.0%

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{k}{\color{blue}{\ell}} \cdot \frac{k}{\ell}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 96.1

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   12. Applied rewrites 96.1%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{\color{blue}{k}}{\ell} \cdot \frac{k}{\ell}\right)} \]

   if 1.15e-101 < k

   1. Initial program 30.4%

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

   2. 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 74.1

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

   4. Applied rewrites 74.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 92.0%

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

Alternative 3: 92.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in 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-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. pow2 N/A

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

   9. frac-times N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

6. Applied rewrites 92.5%

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

Alternative 4: 92.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 0.032:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k\_m \cdot k\_m, 0.08611111111111111\right), k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) (/ (* k_m t) l))))
   (if (<= k_m 0.032)
     (/
      2.0
      (*
       t_1
       (*
        (fma
         (fma
          (fma 0.03432539682539683 (* k_m k_m) 0.08611111111111111)
          (* k_m k_m)
          0.16666666666666666)
         (* k_m k_m)
         1.0)
        (* k_m k_m))))
     (/ 2.0 (* t_1 (/ (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * ((k_m * t) / l);
	double tmp;
	if (k_m <= 0.032) {
		tmp = 2.0 / (t_1 * (fma(fma(fma(0.03432539682539683, (k_m * k_m), 0.08611111111111111), (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m)));
	} else {
		tmp = 2.0 / (t_1 * ((0.5 - (0.5 * cos((2.0 * k_m)))) / cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l))
	tmp = 0.0
	if (k_m <= 0.032)
		tmp = Float64(2.0 / Float64(t_1 * Float64(fma(fma(fma(0.03432539682539683, Float64(k_m * k_m), 0.08611111111111111), Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) / cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 0.032], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(0.03432539682539683 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.032000000000000001

   1. Initial program 41.4%

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

   2. 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 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   9. Applied rewrites 91.9%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.032000000000000001 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 93.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 92.7

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

   8. Applied rewrites 92.7%

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

Alternative 5: 91.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 0.032:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot t\_1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k\_m \cdot k\_m, 0.08611111111111111\right), k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot \left(k\_m \cdot t\_1\right)}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* k_m t) l)))
   (if (<= k_m 0.032)
     (/
      2.0
      (*
       (* (/ k_m l) t_1)
       (*
        (fma
         (fma
          (fma 0.03432539682539683 (* k_m k_m) 0.08611111111111111)
          (* k_m k_m)
          0.16666666666666666)
         (* k_m k_m)
         1.0)
        (* k_m k_m))))
     (/
      2.0
      (/ (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) (* k_m t_1)) (* (cos k_m) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m * t) / l;
	double tmp;
	if (k_m <= 0.032) {
		tmp = 2.0 / (((k_m / l) * t_1) * (fma(fma(fma(0.03432539682539683, (k_m * k_m), 0.08611111111111111), (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m)));
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((2.0 * k_m)))) * (k_m * t_1)) / (cos(k_m) * l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m * t) / l)
	tmp = 0.0
	if (k_m <= 0.032)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * t_1) * Float64(fma(fma(fma(0.03432539682539683, Float64(k_m * k_m), 0.08611111111111111), Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * Float64(k_m * t_1)) / Float64(cos(k_m) * l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.032], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.032000000000000001

   1. Initial program 41.4%

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

   2. 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 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   9. Applied rewrites 91.9%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.032000000000000001 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 91.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 90.7

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

   10. Applied rewrites 90.7%

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

Alternative 6: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.034:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k\_m \cdot k\_m, 0.08611111111111111\right), k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\cos k\_m} \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.034)
   (/
    2.0
    (*
     (* (/ k_m l) (/ (* k_m t) l))
     (*
      (fma
       (fma
        (fma 0.03432539682539683 (* k_m k_m) 0.08611111111111111)
        (* k_m k_m)
        0.16666666666666666)
       (* k_m k_m)
       1.0)
      (* k_m k_m))))
   (/
    2.0
    (*
     (/ (* (* k_m k_m) t) (cos k_m))
     (/ (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.034) {
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (fma(fma(fma(0.03432539682539683, (k_m * k_m), 0.08611111111111111), (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m)));
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / cos(k_m)) * ((0.5 - (0.5 * cos((2.0 * k_m)))) / (l * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.034)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k_m * k_m), 0.08611111111111111), Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) / cos(k_m)) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) / Float64(l * l))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.034], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.034000000000000002

   1. Initial program 41.4%

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

   2. 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 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   9. Applied rewrites 91.9%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.034000000000000002 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 71.0

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

   6. Applied rewrites 71.0%

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

Alternative 7: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.034:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k\_m \cdot k\_m, 0.08611111111111111\right), k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)}{\cos k\_m \cdot \left(\ell \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.034)
   (/
    2.0
    (*
     (* (/ k_m l) (/ (* k_m t) l))
     (*
      (fma
       (fma
        (fma 0.03432539682539683 (* k_m k_m) 0.08611111111111111)
        (* k_m k_m)
        0.16666666666666666)
       (* k_m k_m)
       1.0)
      (* k_m k_m))))
   (/
    2.0
    (/
     (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) (* (* k_m k_m) t))
     (* (cos k_m) (* l l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.034) {
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (fma(fma(fma(0.03432539682539683, (k_m * k_m), 0.08611111111111111), (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m)));
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((2.0 * k_m)))) * ((k_m * k_m) * t)) / (cos(k_m) * (l * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.034)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k_m * k_m), 0.08611111111111111), Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * Float64(Float64(k_m * k_m) * t)) / Float64(cos(k_m) * Float64(l * l))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.034], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.034000000000000002

   1. Initial program 41.4%

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

   2. 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 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   9. Applied rewrites 91.9%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.034000000000000002 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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

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

      4. lift-*.f64 N/A

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

      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. pow2 N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 71.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 71.0

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

   8. Applied rewrites 71.0%

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

Alternative 8: 76.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot t\right)}{\mathsf{fma}\left(-0.5, k\_m \cdot k\_m, 1\right)} \cdot \left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot \frac{k\_m \cdot t}{\ell}\right)}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-101)
   (/
    2.0
    (*
     (/ (* k_m (* k_m t)) (fma -0.5 (* k_m k_m) 1.0))
     (* (/ k_m l) (/ k_m l))))
   (/ 2.0 (/ (* (* k_m k_m) (* k_m (/ (* k_m t) l))) (* (cos k_m) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-101) {
		tmp = 2.0 / (((k_m * (k_m * t)) / fma(-0.5, (k_m * k_m), 1.0)) * ((k_m / l) * (k_m / l)));
	} else {
		tmp = 2.0 / (((k_m * k_m) * (k_m * ((k_m * t) / l))) / (cos(k_m) * l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-101)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m * t)) / fma(-0.5, Float64(k_m * k_m), 1.0)) * Float64(Float64(k_m / l) * Float64(k_m / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(k_m * Float64(Float64(k_m * t) / l))) / Float64(cos(k_m) * l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-101], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-101

   1. Initial program 47.2%

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

   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 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 92.0

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   10. Applied rewrites 92.0%

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{k}{\color{blue}{\ell}} \cdot \frac{k}{\ell}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 96.1

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   12. Applied rewrites 96.1%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \left(\frac{\color{blue}{k}}{\ell} \cdot \frac{k}{\ell}\right)} \]

   if 1.15e-101 < k

   1. Initial program 30.4%

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

   2. 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 74.1

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

   4. Applied rewrites 74.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 92.0%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 66.0

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

   11. Applied rewrites 66.0%

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

Alternative 9: 73.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k\_m \cdot k\_m, 0.08611111111111111\right), k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (/
    2.0
    (*
     (* (/ k_m l) (/ (* k_m t) l))
     (*
      (fma
       (fma
        (fma 0.03432539682539683 (* k_m k_m) 0.08611111111111111)
        (* k_m k_m)
        0.16666666666666666)
       (* k_m k_m)
       1.0)
      (* k_m k_m))))
   (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (fma(fma(fma(0.03432539682539683, (k_m * k_m), 0.08611111111111111), (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m)));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k_m * k_m), 0.08611111111111111), Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 41.3%

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

   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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   9. Applied rewrites 91.6%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 26.5 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 54.9

          \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

   10. Applied rewrites 54.9%

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

Alternative 10: 73.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (/
    2.0
    (*
     (* (/ k_m l) (/ (* k_m t) l))
     (*
      (fma
       (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
       (* k_m k_m)
       1.0)
      (* k_m k_m))))
   (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m)));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 41.3%

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

   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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)} \]

      13. lift-*.f64 91.6

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)} \]

   9. Applied rewrites 91.6%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 26.5 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 54.9

          \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

   10. Applied rewrites 54.9%

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

Alternative 11: 73.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (/
    2.0
    (*
     (* (/ k_m l) (/ (* k_m t) l))
     (* (fma 0.16666666666666666 (* k_m k_m) 1.0) (* k_m k_m))))
   (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (fma(0.16666666666666666, (k_m * k_m), 1.0) * (k_m * k_m)));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l)) * Float64(fma(0.16666666666666666, Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 41.3%

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

   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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 91.5

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

   9. Applied rewrites 91.5%

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

   if 26.5 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 54.9

          \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

   10. Applied rewrites 54.9%

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

Alternative 12: 73.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (/ 2.0 (* (* (/ k_m l) (/ (* k_m t) l)) (* k_m k_m)))
   (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 26.5d0) then
        tmp = 2.0d0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m))
    else
        tmp = ((l * l) / k_m) * ((-0.3333333333333333d0) / (k_m * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 26.5:
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m))
	else:
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(k_m * t) / l)) * Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 26.5)
		tmp = 2.0 / (((k_m / l) * ((k_m * t) / l)) * (k_m * k_m));
	else
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 41.3%

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

   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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 91.3

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

   9. Applied rewrites 91.3%

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

   if 26.5 < k

   1. Initial program 30.9%

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

   2. Taylor expanded in 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 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 54.9

          \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

   10. Applied rewrites 54.9%

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

Alternative 13: 72.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 550000000000:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 550000000000.0)
   (/ 2.0 (* (* (* k_m k_m) t) (* (/ k_m l) (/ k_m l))))
   (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 550000000000.0) {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 550000000000.0d0) then
        tmp = 2.0d0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)))
    else
        tmp = ((l * l) / k_m) * ((-0.3333333333333333d0) / (k_m * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 550000000000.0) {
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 550000000000.0:
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)))
	else:
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 550000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m / l) * Float64(k_m / l))));
	else
		tmp = Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 550000000000.0)
		tmp = 2.0 / (((k_m * k_m) * t) * ((k_m / l) * (k_m / l)));
	else
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 550000000000.0], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e11

   1. Initial program 40.9%

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

   2. Taylor expanded in 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 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 88.9

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 88.4

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

   10. Applied rewrites 88.4%

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

   if 5.5e11 < k

   1. Initial program 31.1%

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

   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 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 55.4

          \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

   10. Applied rewrites 55.4%

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

Alternative 14: 63.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 550000000000:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{k\_m \cdot k\_m} \cdot \frac{2}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 550000000000.0)
   (* (/ (/ (* l l) t) (* k_m k_m)) (/ 2.0 (* k_m k_m)))
   (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 550000000000.0) {
		tmp = (((l * l) / t) / (k_m * k_m)) * (2.0 / (k_m * k_m));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 550000000000.0d0) then
        tmp = (((l * l) / t) / (k_m * k_m)) * (2.0d0 / (k_m * k_m))
    else
        tmp = ((l * l) / k_m) * ((-0.3333333333333333d0) / (k_m * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 550000000000.0) {
		tmp = (((l * l) / t) / (k_m * k_m)) * (2.0 / (k_m * k_m));
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 550000000000.0:
		tmp = (((l * l) / t) / (k_m * k_m)) * (2.0 / (k_m * k_m))
	else:
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 550000000000.0)
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / Float64(k_m * k_m)) * Float64(2.0 / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 550000000000.0)
		tmp = (((l * l) / t) / (k_m * k_m)) * (2.0 / (k_m * k_m));
	else
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 550000000000.0], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e11

   1. Initial program 40.9%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.8

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

   4. Applied rewrites 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 71.3

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

   6. Applied rewrites 71.3%

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

   if 5.5e11 < k

   1. Initial program 31.1%

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

   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 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 55.4

          \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

   10. Applied rewrites 55.4%

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

Alternative 15: 62.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 550000000000:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell \cdot \ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 550000000000.0)
   (* (/ 2.0 (* (* k_m k_m) (* k_m k_m))) (/ (* l l) t))
   (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 550000000000.0) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l * l) / t);
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 550000000000.0d0) then
        tmp = (2.0d0 / ((k_m * k_m) * (k_m * k_m))) * ((l * l) / t)
    else
        tmp = ((l * l) / k_m) * ((-0.3333333333333333d0) / (k_m * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 550000000000.0) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l * l) / t);
	} else {
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 550000000000.0:
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l * l) / t)
	else:
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 550000000000.0)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(k_m * k_m))) * Float64(Float64(l * l) / t));
	else
		tmp = Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 550000000000.0)
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l * l) / t);
	else
		tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 550000000000.0], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e11

   1. Initial program 40.9%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.8

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

   4. Applied rewrites 68.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.8

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

   6. Applied rewrites 68.8%

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

   if 5.5e11 < k

   1. Initial program 31.1%

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

   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 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 55.4

          \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

   10. Applied rewrites 55.4%

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

Alternative 16: 29.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot \ell}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* l l) k_m) (/ -0.3333333333333333 (* k_m t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) / k_m) * ((-0.3333333333333333d0) / (k_m * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) / k_m) * Float64(-0.3333333333333333 / Float64(k_m * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) / k_m) * (-0.3333333333333333 / (k_m * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in 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. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 50.2%

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

8. Taylor expanded in k around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. pow2 N/A

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

   4. associate-*l* N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 29.4

       \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

10. Applied rewrites 29.4%

    \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k \cdot t}} \]
11. Add Preprocessing

Alternative 17: 20.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* -0.11666666666666667 (/ (* l l) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.11666666666666667 * ((l * l) / t);
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (-0.11666666666666667d0) * ((l * l) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.11666666666666667 * ((l * l) / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.11666666666666667 * ((l * l) / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.11666666666666667 * Float64(Float64(l * l) / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.11666666666666667 * ((l * l) / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 28.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 20.5

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

7. Applied rewrites 20.5%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
8. Add Preprocessing

Alternative 18: 18.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.11666666666666667 * (l * (l / t));
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.11666666666666667 * (l * (l / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.11666666666666667 * Float64(l * Float64(l / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.11666666666666667 * (l * (l / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 28.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 20.5

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

7. Applied rewrites 20.5%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 18.0

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

9. Applied rewrites 18.0%

   \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025098 
(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))))