Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 91.8%

Time: 8.6s

Alternatives: 15

Speedup: 5.7×

• 40.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot \left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot t\_1}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 4.8e-9)
     (*
      (/ (/ (fma 0.6666666666666666 (* k_m k_m) 2.0) t) (* k_m k_m))
      (* (/ t_1 k_m) (/ l k_m)))
     (*
      (/ (* 2.0 t_1) (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m))
      (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 4.8e-9) {
		tmp = ((fma(0.6666666666666666, (k_m * k_m), 2.0) / t) / (k_m * k_m)) * ((t_1 / k_m) * (l / k_m));
	} else {
		tmp = ((2.0 * t_1) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m)) * (l / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 4.8e-9)
		tmp = Float64(Float64(Float64(fma(0.6666666666666666, Float64(k_m * k_m), 2.0) / t) / Float64(k_m * k_m)) * Float64(Float64(t_1 / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 * t_1) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m)) * Float64(l / 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[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 4.8e-9], N[(N[(N[(N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * t$95$1), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.8e-9

   1. Initial program 42.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 65.0%

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

   6. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 90.6

           \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   11. Applied rewrites 90.6%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

   if 4.8e-9 < k

   1. Initial program 30.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 91.1

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

   8. Applied rewrites 91.1%

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

   10. Applied rewrites 92.9%

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

Alternative 2: 91.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / (pow(sin(k_m), 2.0) * t)) * (((cos(k_m) * l) / k_m) * (l / 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 / ((sin(k_m) ** 2.0d0) * t)) * (((cos(k_m) * l) / k_m) * (l / k_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 67.9%

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

6. Applied rewrites 67.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 82.8

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

8. Applied rewrites 82.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 91.2

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

10. Applied rewrites 91.2%

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

Alternative 3: 88.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot \left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \frac{\ell}{k\_m}\right) \cdot 2}{k\_m \cdot \left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 4.8e-9)
     (*
      (/ (/ (fma 0.6666666666666666 (* k_m k_m) 2.0) t) (* k_m k_m))
      (* (/ t_1 k_m) (/ l k_m)))
     (/
      (* (* t_1 (/ l k_m)) 2.0)
      (* k_m (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 4.8e-9) {
		tmp = ((fma(0.6666666666666666, (k_m * k_m), 2.0) / t) / (k_m * k_m)) * ((t_1 / k_m) * (l / k_m));
	} else {
		tmp = ((t_1 * (l / k_m)) * 2.0) / (k_m * ((0.5 - (cos((k_m + k_m)) * 0.5)) * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 4.8e-9)
		tmp = Float64(Float64(Float64(fma(0.6666666666666666, Float64(k_m * k_m), 2.0) / t) / Float64(k_m * k_m)) * Float64(Float64(t_1 / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(l / k_m)) * 2.0) / Float64(k_m * Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 4.8e-9], N[(N[(N[(N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.8e-9

   1. Initial program 42.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 65.0%

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

   6. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 90.6

           \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   11. Applied rewrites 90.6%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

   if 4.8e-9 < k

   1. Initial program 30.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 91.1

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

   8. Applied rewrites 91.1%

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

   10. Applied rewrites 86.9%

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

Alternative 4: 84.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ t_2 := \left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\\ \mathbf{if}\;k\_m \leq 0.0025:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot \left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 2.85 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{t\_2} \cdot \left(t\_1 \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot \left(\ell \cdot \ell\right)\right)}{\left(t\_2 \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)) (t_2 (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t)))
   (if (<= k_m 0.0025)
     (*
      (/ (/ (fma 0.6666666666666666 (* k_m k_m) 2.0) t) (* k_m k_m))
      (* (/ t_1 k_m) (/ l k_m)))
     (if (<= k_m 2.85e+150)
       (* (/ 2.0 t_2) (* t_1 (/ l (* k_m k_m))))
       (/ (* 2.0 (* (cos k_m) (* l l))) (* (* t_2 k_m) k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double t_2 = (0.5 - (cos((k_m + k_m)) * 0.5)) * t;
	double tmp;
	if (k_m <= 0.0025) {
		tmp = ((fma(0.6666666666666666, (k_m * k_m), 2.0) / t) / (k_m * k_m)) * ((t_1 / k_m) * (l / k_m));
	} else if (k_m <= 2.85e+150) {
		tmp = (2.0 / t_2) * (t_1 * (l / (k_m * k_m)));
	} else {
		tmp = (2.0 * (cos(k_m) * (l * l))) / ((t_2 * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	t_2 = Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t)
	tmp = 0.0
	if (k_m <= 0.0025)
		tmp = Float64(Float64(Float64(fma(0.6666666666666666, Float64(k_m * k_m), 2.0) / t) / Float64(k_m * k_m)) * Float64(Float64(t_1 / k_m) * Float64(l / k_m)));
	elseif (k_m <= 2.85e+150)
		tmp = Float64(Float64(2.0 / t_2) * Float64(t_1 * Float64(l / Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * Float64(l * l))) / Float64(Float64(t_2 * k_m) * 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[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0025], N[(N[(N[(N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.85e+150], N[(N[(2.0 / t$95$2), $MachinePrecision] * N[(t$95$1 * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.00250000000000000005

   1. Initial program 42.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 65.0%

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

   6. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 73.9

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

   8. Applied rewrites 73.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 90.5

           \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   11. Applied rewrites 90.5%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

   if 0.00250000000000000005 < k < 2.8500000000000001e150

   1. Initial program 21.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 81.6%

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

   6. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if 2.8500000000000001e150 < k

   1. Initial program 38.7%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 69.4%

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

Alternative 5: 82.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0025:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot \left(\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0025)
   (*
    (/ (/ (fma 0.6666666666666666 (* k_m k_m) 2.0) t) (* k_m k_m))
    (* (/ (* (cos k_m) l) k_m) (/ l k_m)))
   (/
    (* 2.0 (* (cos k_m) (* l l)))
    (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0025) {
		tmp = ((fma(0.6666666666666666, (k_m * k_m), 2.0) / t) / (k_m * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m));
	} else {
		tmp = (2.0 * (cos(k_m) * (l * l))) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0025)
		tmp = Float64(Float64(Float64(fma(0.6666666666666666, Float64(k_m * k_m), 2.0) / t) / Float64(k_m * k_m)) * Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * Float64(l * l))) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0025], N[(N[(N[(N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 42.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 65.0%

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

   6. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 73.9

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

   8. Applied rewrites 73.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 90.5

           \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   11. Applied rewrites 90.5%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

   if 0.00250000000000000005 < k

   1. Initial program 30.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 75.5%

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

Alternative 6: 82.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 0.0025:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot \left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 0.0025)
     (*
      (/ (/ (fma 0.6666666666666666 (* k_m k_m) 2.0) t) (* k_m k_m))
      (* (/ t_1 k_m) (/ l k_m)))
     (/
      (* (* t_1 l) 2.0)
      (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 0.0025) {
		tmp = ((fma(0.6666666666666666, (k_m * k_m), 2.0) / t) / (k_m * k_m)) * ((t_1 / k_m) * (l / k_m));
	} else {
		tmp = ((t_1 * l) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 0.0025)
		tmp = Float64(Float64(Float64(fma(0.6666666666666666, Float64(k_m * k_m), 2.0) / t) / Float64(k_m * k_m)) * Float64(Float64(t_1 / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(t_1 * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * 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[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0025], N[(N[(N[(N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 42.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 65.0%

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

   6. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 73.9

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

   8. Applied rewrites 73.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 90.5

           \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   11. Applied rewrites 90.5%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

   if 0.00250000000000000005 < k

   1. Initial program 30.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 70.7%

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

   6. Applied rewrites 75.5%

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

Alternative 7: 73.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 67.9%

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

6. Applied rewrites 67.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 82.8

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

8. Applied rewrites 82.8%

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

9. Taylor expanded in k around 0

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

    1. count-2-rev N/A

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

    2. *-commutative N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 73.3

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

11. Applied rewrites 73.3%

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

Alternative 8: 70.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 1.55 \cdot 10^{-111}:\\ \;\;\;\;\frac{2}{\left(0.5 - 0.5\right) \cdot t} \cdot \left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{t\_1 \cdot \ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 1.55e-111)
     (* (/ 2.0 (* (- 0.5 0.5) t)) (* (/ t_1 k_m) (/ l k_m)))
     (* (/ 2.0 (* (* k_m k_m) t)) (/ (* t_1 l) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 1.55e-111) {
		tmp = (2.0 / ((0.5 - 0.5) * t)) * ((t_1 / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((t_1 * l) / (k_m * k_m));
	}
	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 = cos(k_m) * l
    if (k_m <= 1.55d-111) then
        tmp = (2.0d0 / ((0.5d0 - 0.5d0) * t)) * ((t_1 / k_m) * (l / k_m))
    else
        tmp = (2.0d0 / ((k_m * k_m) * t)) * ((t_1 * l) / (k_m * k_m))
    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.cos(k_m) * l;
	double tmp;
	if (k_m <= 1.55e-111) {
		tmp = (2.0 / ((0.5 - 0.5) * t)) * ((t_1 / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((t_1 * l) / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if k_m <= 1.55e-111:
		tmp = (2.0 / ((0.5 - 0.5) * t)) * ((t_1 / k_m) * (l / k_m))
	else:
		tmp = (2.0 / ((k_m * k_m) * t)) * ((t_1 * l) / (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 1.55e-111)
		tmp = Float64(Float64(2.0 / Float64(Float64(0.5 - 0.5) * t)) * Float64(Float64(t_1 / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(t_1 * l) / Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (k_m <= 1.55e-111)
		tmp = (2.0 / ((0.5 - 0.5) * t)) * ((t_1 / k_m) * (l / k_m));
	else
		tmp = (2.0 / ((k_m * k_m) * t)) * ((t_1 * l) / (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.55e-111], N[(N[(2.0 / N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000007e-111

   1. Initial program 48.9%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 74.4%

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

   6. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 87.9

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

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in k around 0

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

       1. count-2-rev 87.9

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

       2. *-commutative 87.9

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

   11. Applied rewrites 87.9%

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

   if 1.55000000000000007e-111 < k

   1. Initial program 29.9%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 64.8%

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

   6. Applied rewrites 64.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 62.5

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

   9. Applied rewrites 62.5%

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

Alternative 9: 66.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m} \cdot \left(\frac{\ell}{t} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\left(\cos k\_m \cdot \ell\right) \cdot \ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1.25e-161)
   (* (/ 2.0 (* (* (* k_m k_m) k_m) k_m)) (* (/ l t) l))
   (* (/ 2.0 (* (* k_m k_m) t)) (/ (* (* (cos k_m) l) l) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.25e-161) {
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l);
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * (((cos(k_m) * l) * l) / (k_m * k_m));
	}
	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 (l <= 1.25d-161) then
        tmp = (2.0d0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l)
    else
        tmp = (2.0d0 / ((k_m * k_m) * t)) * (((cos(k_m) * l) * l) / (k_m * k_m))
    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 (l <= 1.25e-161) {
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l);
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * (((Math.cos(k_m) * l) * l) / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1.25e-161:
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l)
	else:
		tmp = (2.0 / ((k_m * k_m) * t)) * (((math.cos(k_m) * l) * l) / (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 1.25e-161)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * k_m) * k_m)) * Float64(Float64(l / t) * l));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(Float64(cos(k_m) * l) * l) / Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 1.25e-161)
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l);
	else
		tmp = (2.0 / ((k_m * k_m) * t)) * (((cos(k_m) * l) * l) / (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1.25e-161], N[(N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.25e-161

   1. Initial program 33.4%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 60.7

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

   4. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 60.5

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

   6. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 64.6%

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

   if 1.25e-161 < l

   1. Initial program 40.3%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 74.0%

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

   6. Applied rewrites 74.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 70.6

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

   9. Applied rewrites 70.6%

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

Alternative 10: 66.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m} \cdot \left(\frac{\ell}{t} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.6e+15)
   (* (/ 2.0 (* (* (* k_m k_m) k_m) k_m)) (* (/ l t) l))
   (/ (* 2.0 (* l l)) (* (* k_m k_m) (* (* k_m k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.6e+15) {
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l);
	} else {
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * ((k_m * 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 (t <= 1.6d+15) then
        tmp = (2.0d0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l)
    else
        tmp = (2.0d0 * (l * l)) / ((k_m * k_m) * ((k_m * 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 (t <= 1.6e+15) {
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l);
	} else {
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * ((k_m * k_m) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.6e+15:
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l)
	else:
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * ((k_m * k_m) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.6e+15)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * k_m) * k_m)) * Float64(Float64(l / t) * l));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(k_m * k_m) * Float64(Float64(k_m * k_m) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.6e+15)
		tmp = (2.0 / (((k_m * k_m) * k_m) * k_m)) * ((l / t) * l);
	else
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * ((k_m * k_m) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.6e+15], N[(N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.6e15

   1. Initial program 39.3%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 60.7

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

   4. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 60.5

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

   6. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 64.3%

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

   if 1.6e15 < t

   1. Initial program 25.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.8

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

   4. Applied rewrites 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 71.7

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

   6. Applied rewrites 71.7%

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

Alternative 11: 64.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * (l * l)) / ((k_m * k_m) * ((k_m * 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 = (2.0d0 * (l * l)) / ((k_m * k_m) * ((k_m * k_m) * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.6

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

4. Applied rewrites 62.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. unpow-prod-down N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 64.1

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

6. Applied rewrites 64.1%

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

Alternative 12: 29.2% accurate, 7.8× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / ((k_m * 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) * (-0.3333333333333333d0)) / ((k_m * 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) * -0.3333333333333333) / ((k_m * k_m) * t);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 67.9%

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

5. 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 29.8%

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

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. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 29.2

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

10. Applied rewrites 29.2%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
11. Add Preprocessing

Alternative 13: 20.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (/ (* (* l l) -0.11666666666666667) t))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * -0.11666666666666667) / 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) * (-0.11666666666666667d0)) / t
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) * -0.11666666666666667) / t;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) * -0.11666666666666667) / t

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) * -0.11666666666666667) / t)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) * -0.11666666666666667) / t;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] * -0.11666666666666667), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in 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 28.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334\right) \cdot \left(k \cdot k\right), -2, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \frac{-7}{60} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \frac{-7}{60} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   5. lift-/.f64 20.6

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \]

7. Applied rewrites 20.6%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   4. pow2 N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \frac{-7}{60} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-7}{60}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-7}{60}}{t} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-7}{60}}{t} \]

   8. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-7}{60}}{t} \]

   9. lift-*.f64 20.6

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t} \]

9. Applied rewrites 20.6%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t} \]
10. Add Preprocessing

Alternative 14: 20.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (/ (* l l) t) -0.11666666666666667))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) / t) * -0.11666666666666667;
}

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) / t) * (-0.11666666666666667d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) / t) * -0.11666666666666667;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) / t) * -0.11666666666666667

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) / t) * -0.11666666666666667)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) / t) * -0.11666666666666667;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.11666666666666667), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in 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 28.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334\right) \cdot \left(k \cdot k\right), -2, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \frac{-7}{60} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \frac{-7}{60} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   5. lift-/.f64 20.6

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \]

7. Applied rewrites 20.6%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
8. Add Preprocessing

Alternative 15: 18.7% accurate, 12.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* l (* (/ l t) -0.11666666666666667)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * ((l / t) * -0.11666666666666667);
}

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 / t) * (-0.11666666666666667d0))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * ((l / t) * -0.11666666666666667);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * ((l / t) * -0.11666666666666667)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(Float64(l / t) * -0.11666666666666667))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * ((l / t) * -0.11666666666666667);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(N[(l / t), $MachinePrecision] * -0.11666666666666667), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in 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 28.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334\right) \cdot \left(k \cdot k\right), -2, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \frac{-7}{60} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \frac{-7}{60} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   5. lift-/.f64 20.6

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \]

7. Applied rewrites 20.6%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \frac{-7}{60} \]

   3. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-7}{60} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-7}{60} \]

   5. lower-/.f64 18.7

      \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.11666666666666667 \]

9. Applied rewrites 18.7%

   \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.11666666666666667 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-7}{60} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-7}{60} \]

    3. lift-/.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-7}{60} \]

    4. associate-*l* N/A

       \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{-7}{60}}\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{-7}{60}}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot \frac{-7}{60}\right) \]

    7. lift-/.f64 18.7

       \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \]

11. Applied rewrites 18.7%

    \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot \color{blue}{-0.11666666666666667}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(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))))