Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 97.4%

Time: 9.6s

Alternatives: 22

Speedup: 9.6×

• 39.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((((t ** (-1.0d0)) / sin(k)) * (2.0d0 / sin(k))) * ((cos(k) * l) / k)) * (l / k)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 73.4

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

5. Applied rewrites 73.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. frac-times N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. pow2 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 90.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.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. lift-/.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

9. Applied rewrites 96.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-pow.f64 N/A

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

    4. lift-sin.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-/r* N/A

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

    7. metadata-eval N/A

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

    8. associate-*r/ N/A

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

    9. *-commutative N/A

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

    10. unpow2 N/A

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

    11. times-frac N/A

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

    12. lower-*.f64 N/A

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

    13. lower-/.f64 N/A

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

    14. inv-pow N/A

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

    15. lower-pow.f64 N/A

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

    16. lift-sin.f64 N/A

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

    17. lower-/.f64 N/A

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

    18. lift-sin.f64 97.4

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

11. Applied rewrites 97.4%

    \[\leadsto \left(\left(\frac{{t}^{-1}}{\sin k} \cdot \frac{2}{\sin k}\right) \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
12. Add Preprocessing

Alternative 2: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 50000000000000:\\ \;\;\;\;\left(\frac{2}{k \cdot t} \cdot \frac{t\_1}{{\sin k}^{2}}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{t\_1}{k}\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= k 50000000000000.0)
     (* (* (/ 2.0 (* k t)) (/ t_1 (pow (sin k) 2.0))) (/ l k))
     (* (* (/ 2.0 (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)) (/ t_1 k)) (/ l k)))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (k <= 50000000000000.0) {
		tmp = ((2.0 / (k * t)) * (t_1 / pow(sin(k), 2.0))) * (l / k);
	} else {
		tmp = ((2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l
    if (k <= 50000000000000.0d0) then
        tmp = ((2.0d0 / (k * t)) * (t_1 / (sin(k) ** 2.0d0))) * (l / k)
    else
        tmp = ((2.0d0 / ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * t)) * (t_1 / k)) * (l / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(k) * l;
	double tmp;
	if (k <= 50000000000000.0) {
		tmp = ((2.0 / (k * t)) * (t_1 / Math.pow(Math.sin(k), 2.0))) * (l / k);
	} else {
		tmp = ((2.0 / ((0.5 - (0.5 * Math.cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(k) * l
	tmp = 0
	if k <= 50000000000000.0:
		tmp = ((2.0 / (k * t)) * (t_1 / math.pow(math.sin(k), 2.0))) * (l / k)
	else:
		tmp = ((2.0 / ((0.5 - (0.5 * math.cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (k <= 50000000000000.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(k * t)) * Float64(t_1 / (sin(k) ^ 2.0))) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t)) * Float64(t_1 / k)) * Float64(l / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(k) * l;
	tmp = 0.0;
	if (k <= 50000000000000.0)
		tmp = ((2.0 / (k * t)) * (t_1 / (sin(k) ^ 2.0))) * (l / k);
	else
		tmp = ((2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k, 50000000000000.0], N[(N[(N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5e13

   1. Initial program 36.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.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. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 95.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-cos.f64 N/A

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

       9. frac-times N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. times-frac N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-/.f64 N/A

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

   11. Applied rewrites 94.1%

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

   if 5e13 < k

   1. Initial program 31.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.4

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

   5. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.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. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 99.5%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 99.2

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

   11. Applied rewrites 99.2%

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

Alternative 3: 89.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{t\_1}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{t\_1}{k}\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= k 3e+16)
     (* (* (/ t_1 (* (* k k) t)) (/ l (pow (sin k) 2.0))) 2.0)
     (* (* (/ 2.0 (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)) (/ t_1 k)) (/ l k)))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (k <= 3e+16) {
		tmp = ((t_1 / ((k * k) * t)) * (l / pow(sin(k), 2.0))) * 2.0;
	} else {
		tmp = ((2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k) * l
    if (k <= 3d+16) then
        tmp = ((t_1 / ((k * k) * t)) * (l / (sin(k) ** 2.0d0))) * 2.0d0
    else
        tmp = ((2.0d0 / ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * t)) * (t_1 / k)) * (l / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(k) * l;
	double tmp;
	if (k <= 3e+16) {
		tmp = ((t_1 / ((k * k) * t)) * (l / Math.pow(Math.sin(k), 2.0))) * 2.0;
	} else {
		tmp = ((2.0 / ((0.5 - (0.5 * Math.cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(k) * l
	tmp = 0
	if k <= 3e+16:
		tmp = ((t_1 / ((k * k) * t)) * (l / math.pow(math.sin(k), 2.0))) * 2.0
	else:
		tmp = ((2.0 / ((0.5 - (0.5 * math.cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (k <= 3e+16)
		tmp = Float64(Float64(Float64(t_1 / Float64(Float64(k * k) * t)) * Float64(l / (sin(k) ^ 2.0))) * 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t)) * Float64(t_1 / k)) * Float64(l / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(k) * l;
	tmp = 0.0;
	if (k <= 3e+16)
		tmp = ((t_1 / ((k * k) * t)) * (l / (sin(k) ^ 2.0))) * 2.0;
	else
		tmp = ((2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * t)) * (t_1 / k)) * (l / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k, 3e+16], N[(N[(N[(t$95$1 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3e16

   1. Initial program 36.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.7

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

   5. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 87.0%

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

   if 3e16 < k

   1. Initial program 31.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.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. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 99.5%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 99.2

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

   11. Applied rewrites 99.2%

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

Alternative 4: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\cos k \cdot \ell}{k} \cdot \frac{\frac{\ell}{k} \cdot 2}{{\sin k}^{2} \cdot t} \end{array} \]

FPCore

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

C

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

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 = ((cos(k) * l) / k) * (((l / k) * 2.0d0) / ((sin(k) ** 2.0d0) * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 73.4

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

5. Applied rewrites 73.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. frac-times N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. pow2 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 90.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.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. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l/ N/A

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

   13. frac-times N/A

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

   14. lower-/.f64 N/A

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

9. Applied rewrites 86.6%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-cos.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. associate-*l* N/A

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

    8. lift-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-pow.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. times-frac N/A

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

    13. lower-*.f64 N/A

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

    14. lift-cos.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. lift-/.f64 N/A

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

    17. lower-/.f64 N/A

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

11. Applied rewrites 96.3%

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

Alternative 5: 96.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 / ((sin(k) ** 2.0d0) * t)) * ((cos(k) * l) / k)) * (l / k)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 73.4

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

5. Applied rewrites 73.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. frac-times N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. pow2 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 90.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.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. lift-/.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

9. Applied rewrites 96.1%

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

Alternative 6: 71.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00165:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+176}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.00165)
   (*
    (*
     (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k))
     (/ (* (cos k) l) k))
    (/ l k))
   (if (<= k 2.8e+176)
     (*
      (/ 2.0 (* (* k k) t))
      (/ (* (cos k) (* l l)) (- 0.5 (* 0.5 (cos (* 2.0 k))))))
     (* (/ 2.0 (* (pow (sin k) 2.0) t)) (* (/ l k) (/ l k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.00165) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * ((cos(k) * l) / k)) * (l / k);
	} else if (k <= 2.8e+176) {
		tmp = (2.0 / ((k * k) * t)) * ((cos(k) * (l * l)) / (0.5 - (0.5 * cos((2.0 * k)))));
	} else {
		tmp = (2.0 / (pow(sin(k), 2.0) * t)) * ((l / k) * (l / k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.00165)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k * k), 2.0) / t) / Float64(k * k)) * Float64(Float64(cos(k) * l) / k)) * Float64(l / k));
	elseif (k <= 2.8e+176)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(cos(k) * Float64(l * l)) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))));
	else
		tmp = Float64(Float64(2.0 / Float64((sin(k) ^ 2.0) * t)) * Float64(Float64(l / k) * Float64(l / k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.00165], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.8e+176], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.00165

   1. Initial program 36.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.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. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 94.9%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lower-*.f64 70.5

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

   12. Applied rewrites 70.5%

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

   if 0.00165 < k < 2.8000000000000002e176

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 77.1

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

   7. Applied rewrites 77.1%

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

   if 2.8000000000000002e176 < k

   1. Initial program 38.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 60.9

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

   5. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 66.2%

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

Alternative 7: 71.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 0.00165:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{t\_1}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{t\_1 \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= k 0.00165)
     (*
      (* (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k)) (/ t_1 k))
      (/ l k))
     (if (<= k 4.6e+154)
       (* (/ 2.0 (* k k)) (/ (* t_1 l) (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)))
       (* (/ 2.0 (* (pow (sin k) 2.0) t)) (* (/ l k) (/ l k)))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (k <= 0.00165) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * (t_1 / k)) * (l / k);
	} else if (k <= 4.6e+154) {
		tmp = (2.0 / (k * k)) * ((t_1 * l) / ((0.5 - (0.5 * cos((2.0 * k)))) * t));
	} else {
		tmp = (2.0 / (pow(sin(k), 2.0) * t)) * ((l / k) * (l / k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (k <= 0.00165)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k * k), 2.0) / t) / Float64(k * k)) * Float64(t_1 / k)) * Float64(l / k));
	elseif (k <= 4.6e+154)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(t_1 * l) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t)));
	else
		tmp = Float64(Float64(2.0 / Float64((sin(k) ^ 2.0) * t)) * Float64(Float64(l / k) * Float64(l / k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k, 0.00165], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.6e+154], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * l), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.00165

   1. Initial program 36.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.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. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 94.9%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lower-*.f64 70.5

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

   12. Applied rewrites 70.5%

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

   if 0.00165 < k < 4.6e154

   1. Initial program 24.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 81.4

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

   5. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 81.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 81.4

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

   9. Applied rewrites 81.4%

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

   if 4.6e154 < k

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 58.9

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

   5. Applied rewrites 58.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 64.3%

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

Alternative 8: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos k \cdot \ell}{k}\\ \mathbf{if}\;k \leq 0.00162:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot t\_1\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot t\_1\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* (cos k) l) k)))
   (if (<= k 0.00162)
     (* (* (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k)) t_1) (/ l k))
     (* (* (/ 2.0 (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)) t_1) (/ l k)))))

C

double code(double t, double l, double k) {
	double t_1 = (cos(k) * l) / k;
	double tmp;
	if (k <= 0.00162) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * t_1) * (l / k);
	} else {
		tmp = ((2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * t)) * t_1) * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(cos(k) * l) / k)
	tmp = 0.0
	if (k <= 0.00162)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k * k), 2.0) / t) / Float64(k * k)) * t_1) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t)) * t_1) * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[k, 0.00162], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0016199999999999999

   1. Initial program 36.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.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. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 94.9%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lower-*.f64 70.5

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

   12. Applied rewrites 70.5%

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

   if 0.0016199999999999999 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 91.3%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   9. Applied rewrites 99.5%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       2. lift-sin.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{2}{\left(\sin k \cdot \sin k\right) \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       4. sqr-sin-a N/A

          \[\leadsto \left(\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       5. lower--.f64 N/A

          \[\leadsto \left(\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       7. lower-cos.f64 N/A

          \[\leadsto \left(\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       8. lower-*.f64 99.1

          \[\leadsto \left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   11. Applied rewrites 99.1%

       \[\leadsto \left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 0.00165:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{t\_1}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= k 0.00165)
     (*
      (* (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k)) (/ t_1 k))
      (/ l k))
     (/ (* (* t_1 (/ l k)) 2.0) (* k (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (k <= 0.00165) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * (t_1 / k)) * (l / k);
	} else {
		tmp = ((t_1 * (l / k)) * 2.0) / (k * ((0.5 - (0.5 * cos((2.0 * k)))) * t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (k <= 0.00165)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k * k), 2.0) / t) / Float64(k * k)) * Float64(t_1 / k)) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(l / k)) * 2.0) / Float64(k * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k, 0.00165], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00165

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 74.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 90.7%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   9. Applied rewrites 94.9%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       2. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       3. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2 \cdot 1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       4. metadata-eval N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       5. div-add-rev N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       6. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, {k}^{2}, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       8. pow2 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       11. lower-*.f64 70.5

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   12. Applied rewrites 70.5%

       \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   if 0.00165 < k

   1. Initial program 30.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 70.1

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 91.3%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{2}{{\sin k}^{2} \cdot t}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}}{k} \cdot \frac{\color{blue}{2}}{{\sin k}^{2} \cdot t} \]

      13. frac-times N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   9. Applied rewrites 87.6%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]

       2. lift-sin.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(\sin k \cdot \sin k\right) \cdot t\right)} \]

       4. sqr-sin-a N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right)} \]

       7. lower-cos.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right)} \]

       8. lower-*.f64 87.3

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right)} \]

   11. Applied rewrites 87.3%

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.1e+102)
   (*
    (*
     (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k))
     (/ (* (cos k) l) k))
    (/ l k))
   (* (* (/ 2.0 (* (pow (sin k) 2.0) t)) (/ l k)) (/ l k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e+102) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * ((cos(k) * l) / k)) * (l / k);
	} else {
		tmp = ((2.0 / (pow(sin(k), 2.0) * t)) * (l / k)) * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.1e+102)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k * k), 2.0) / t) / Float64(k * k)) * Float64(Float64(cos(k) * l) / k)) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64((sin(k) ^ 2.0) * t)) * Float64(l / k)) * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.1e+102], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.09999999999999987e102

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 75.4

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 90.4%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   9. Applied rewrites 95.4%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       2. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       3. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2 \cdot 1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       4. metadata-eval N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       5. div-add-rev N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       6. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, {k}^{2}, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       8. pow2 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       11. lower-*.f64 69.0

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   12. Applied rewrites 69.0%

       \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   if 3.09999999999999987e102 < k

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 63.4

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 93.1%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   9. Applied rewrites 99.5%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.5%

       \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left({\sin k}^{2} \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.1e+102)
   (*
    (*
     (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k))
     (/ (* (cos k) l) k))
    (/ l k))
   (/ (* (* l (/ l k)) 2.0) (* k (* (pow (sin k) 2.0) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e+102) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * ((cos(k) * l) / k)) * (l / k);
	} else {
		tmp = ((l * (l / k)) * 2.0) / (k * (pow(sin(k), 2.0) * t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.1e+102)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k * k), 2.0) / t) / Float64(k * k)) * Float64(Float64(cos(k) * l) / k)) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(l * Float64(l / k)) * 2.0) / Float64(k * Float64((sin(k) ^ 2.0) * t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.1e+102], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.09999999999999987e102

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 75.4

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 90.4%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   9. Applied rewrites 95.4%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       2. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       3. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2 \cdot 1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       4. metadata-eval N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       5. div-add-rev N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       6. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, {k}^{2}, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       8. pow2 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       11. lower-*.f64 69.0

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   12. Applied rewrites 69.0%

       \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   if 3.09999999999999987e102 < k

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 63.4

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 93.1%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{2}{{\sin k}^{2} \cdot t}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}}{k} \cdot \frac{\color{blue}{2}}{{\sin k}^{2} \cdot t} \]

      13. frac-times N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   9. Applied rewrites 88.2%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 61.5%

       \[\leadsto \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left({\sin k}^{2} \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 1.8 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{t\_1}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= k 1.8e+130)
     (*
      (* (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k)) (/ t_1 k))
      (/ l k))
     (/ (* (* t_1 (/ l k)) 2.0) (* k (* (* k k) t))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double tmp;
	if (k <= 1.8e+130) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * (t_1 / k)) * (l / k);
	} else {
		tmp = ((t_1 * (l / k)) * 2.0) / (k * ((k * k) * t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	tmp = 0.0
	if (k <= 1.8e+130)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k * k), 2.0) / t) / Float64(k * k)) * Float64(t_1 / k)) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(l / k)) * 2.0) / Float64(k * Float64(Float64(k * k) * t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k, 1.8e+130], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8000000000000001e130

   1. Initial program 35.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 75.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 90.5%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   9. Applied rewrites 95.5%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       2. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       3. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2 \cdot 1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       4. metadata-eval N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       5. div-add-rev N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       6. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, {k}^{2}, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       8. pow2 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       11. lower-*.f64 68.6

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   12. Applied rewrites 68.6%

       \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   if 1.8000000000000001e130 < k

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 60.9

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 93.2%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.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. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{2}{{\sin k}^{2} \cdot t}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}}{k} \cdot \frac{\color{blue}{2}}{{\sin k}^{2} \cdot t} \]

      13. frac-times N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   9. Applied rewrites 88.2%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left({k}^{2} \cdot t\right)} \]
   11. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

       2. lower-*.f64 59.9

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. Applied rewrites 59.9%

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 72.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{k \cdot k}}{t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (/ 2.0 (* k k)) t) (* (/ (* (cos k) l) k) (/ l k))))

C

double code(double t, double l, double k) {
	return ((2.0 / (k * k)) / t) * (((cos(k) * l) / k) * (l / k));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((2.0d0 / (k * k)) / t) * (((cos(k) * l) / k) * (l / k))
end function

Java

public static double code(double t, double l, double k) {
	return ((2.0 / (k * k)) / t) * (((Math.cos(k) * l) / k) * (l / k));
}

Python

def code(t, l, k):
	return ((2.0 / (k * k)) / t) * (((math.cos(k) * l) / k) * (l / k))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(2.0 / Float64(k * k)) / t) * Float64(Float64(Float64(cos(k) * l) / k) * Float64(l / k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((2.0 / (k * k)) / t) * (((cos(k) * l) / k) * (l / k));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 73.4

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. frac-times N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

   14. pow2 N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

   15. associate-*r* N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   16. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

7. Applied rewrites 90.9%

   \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]
9. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{{k}^{2}}}{t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

   2. pow2 N/A

      \[\leadsto \frac{\frac{2}{k \cdot k}}{t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{k \cdot k}}{t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{2}{k \cdot k}}{t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   5. lift-*.f64 72.5

      \[\leadsto \frac{\frac{2}{k \cdot k}}{t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

10. Applied rewrites 72.5%

    \[\leadsto \frac{\frac{2}{k \cdot k}}{t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]
11. Add Preprocessing

Alternative 14: 70.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* (* (* (cos k) l) (/ l k)) 2.0) (* k (* (* k k) t))))

C

double code(double t, double l, double k) {
	return (((cos(k) * l) * (l / k)) * 2.0) / (k * ((k * k) * t));
}

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 = (((cos(k) * l) * (l / k)) * 2.0d0) / (k * ((k * k) * t))
end function

Java

public static double code(double t, double l, double k) {
	return (((Math.cos(k) * l) * (l / k)) * 2.0) / (k * ((k * k) * t));
}

Python

def code(t, l, k):
	return (((math.cos(k) * l) * (l / k)) * 2.0) / (k * ((k * k) * t))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(Float64(cos(k) * l) * Float64(l / k)) * 2.0) / Float64(k * Float64(Float64(k * k) * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (((cos(k) * l) * (l / k)) * 2.0) / (k * ((k * k) * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 73.4

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. frac-times N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

   14. pow2 N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

   15. associate-*r* N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   16. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

7. Applied rewrites 90.9%

   \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   5. lift-sin.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   9. lift-cos.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. lift-/.f64 N/A

       \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   11. *-commutative N/A

       \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{2}{{\sin k}^{2} \cdot t}} \]

   12. associate-*l/ N/A

       \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}}{k} \cdot \frac{\color{blue}{2}}{{\sin k}^{2} \cdot t} \]

   13. frac-times N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

9. Applied rewrites 86.6%

   \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{\color{blue}{k \cdot \left({\sin k}^{2} \cdot t\right)}} \]

10. Taylor expanded in k around 0

    \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left({k}^{2} \cdot t\right)} \]
11. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

    2. lower-*.f64 70.8

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

12. Applied rewrites 70.8%

    \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k}\right) \cdot 2}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]
13. Add Preprocessing

Alternative 15: 71.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \ell}{k \cdot k} \cdot \frac{2}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.4e-90)
   (* (/ 2.0 (* (* k k) t)) (* (/ l k) (/ l k)))
   (* (/ (* (/ l t) l) (* k k)) (/ 2.0 (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.4e-90) {
		tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
	} else {
		tmp = (((l / t) * l) / (k * k)) * (2.0 / (k * k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.4d-90) then
        tmp = (2.0d0 / ((k * k) * t)) * ((l / k) * (l / k))
    else
        tmp = (((l / t) * l) / (k * k)) * (2.0d0 / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.4e-90) {
		tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
	} else {
		tmp = (((l / t) * l) / (k * k)) * (2.0 / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.4e-90:
		tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k))
	else:
		tmp = (((l / t) * l) / (k * k)) * (2.0 / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.4e-90)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l / k) * Float64(l / k)));
	else
		tmp = Float64(Float64(Float64(Float64(l / t) * l) / Float64(k * k)) * Float64(2.0 / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.4e-90)
		tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
	else
		tmp = (((l / t) * l) / (k * k)) * (2.0 / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.4e-90], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.39999999999999994e-90

   1. Initial program 38.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 73.9

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lower-/.f64 76.3

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied rewrites 76.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

   if 3.39999999999999994e-90 < k

   1. Initial program 29.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-*.f64 57.0

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

      5. lower-/.f64 59.4

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   7. Applied rewrites 59.4%

      \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\color{blue}{\ell} \cdot \frac{\ell}{t}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2 \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{{k}^{4}}} \]

      8. pow2 N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      10. pow-prod-up N/A

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t} \cdot 2}{\color{blue}{{k}^{2}} \cdot {k}^{2}} \]

      12. times-frac N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot \color{blue}{\frac{2}{{k}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot \color{blue}{\frac{2}{{k}^{2}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot \frac{\color{blue}{2}}{{k}^{2}} \]

      15. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      16. associate-*r/ N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      19. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{{k}^{2}} \cdot \frac{2}{{k}^{2}} \]

      20. pow2 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k \cdot k} \cdot \frac{2}{{k}^{2}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k \cdot k} \cdot \frac{2}{{k}^{2}} \]

      22. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2}}} \]

      23. pow2 N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k \cdot k} \cdot \frac{2}{k \cdot \color{blue}{k}} \]

   9. Applied rewrites 61.4%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{k \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 64.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.5e-172)
   (* (/ 2.0 (* (* k k) (* k k))) (* l (/ l t)))
   (* (/ 2.0 (* k k)) (/ (* l l) (* (* k k) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.5e-172) {
		tmp = (2.0 / ((k * k) * (k * k))) * (l * (l / t));
	} else {
		tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.5d-172) then
        tmp = (2.0d0 / ((k * k) * (k * k))) * (l * (l / t))
    else
        tmp = (2.0d0 / (k * k)) * ((l * l) / ((k * k) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.5e-172) {
		tmp = (2.0 / ((k * k) * (k * k))) * (l * (l / t));
	} else {
		tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.5e-172:
		tmp = (2.0 / ((k * k) * (k * k))) * (l * (l / t))
	else:
		tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.5e-172)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(l * Float64(l / t)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l * l) / Float64(Float64(k * k) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.5e-172)
		tmp = (2.0 / ((k * k) * (k * k))) * (l * (l / t));
	else
		tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.5e-172], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.49999999999999992e-172

   1. Initial program 32.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-*.f64 58.9

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

      5. lower-/.f64 63.4

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   7. Applied rewrites 63.4%

      \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{{k}^{\left(2 + 2\right)}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      3. pow-prod-up N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

      8. lower-*.f64 63.3

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

   9. Applied rewrites 63.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

   if 1.49999999999999992e-172 < l

   1. Initial program 39.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 77.7

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   7. Applied rewrites 77.9%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2} \cdot t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 67.0

         \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

   10. Applied rewrites 67.0%

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 71.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* (* k k) t)) (* (/ l k) (/ l k))))

C

double code(double t, double l, double k) {
	return (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 / ((k * k) * t)) * ((l / k) * (l / k))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
}

Python

def code(t, l, k):
	return (2.0 / ((k * k) * t)) * ((l / k) * (l / k))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l / k) * Float64(l / k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 73.4

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
7. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   6. lower-/.f64 71.0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

8. Applied rewrites 71.0%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]
9. Add Preprocessing

Alternative 18: 64.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* k k)) (/ (* l l) (* (* k k) t))))

C

double code(double t, double l, double k) {
	return (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 / (k * k)) * ((l * l) / ((k * k) * t))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
}

Python

def code(t, l, k):
	return (2.0 / (k * k)) * ((l * l) / ((k * k) * t))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l * l) / Float64(Float64(k * k) * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 73.4

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. frac-times N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

   14. pow2 N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

   15. associate-*r* N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

7. Applied rewrites 73.4%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2} \cdot t}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

   6. lift-*.f64 64.6

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

10. Applied rewrites 64.6%

    \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
11. Add Preprocessing

Alternative 19: 29.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \ell\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ -0.3333333333333333 (* k k)) (* (/ l t) l)))

C

double code(double t, double l, double k) {
	return (-0.3333333333333333 / (k * k)) * ((l / t) * l);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((-0.3333333333333333d0) / (k * k)) * ((l / t) * l)
end function

Java

public static double code(double t, double l, double k) {
	return (-0.3333333333333333 / (k * k)) * ((l / t) * l);
}

Python

def code(t, l, k):
	return (-0.3333333333333333 / (k * k)) * ((l / t) * l)

Julia

function code(t, l, k)
	return Float64(Float64(-0.3333333333333333 / Float64(k * k)) * Float64(Float64(l / t) * l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (-0.3333333333333333 / (k * k)) * ((l / t) * l);
end

Wolfram

code[t_, l_, k_] := N[(N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 28.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 20.4

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 20.4%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

11. Applied rewrites 46.9%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{{\left(\ell \cdot k\right)}^{2}}{t}, -0.3333333333333333, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\color{blue}{{k}^{4}}} \]

12. Taylor expanded in k around inf

    \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

    2. times-frac N/A

       \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{\frac{-1}{3}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t} \]

    5. pow2 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{{\ell}^{2}}{t} \]

    7. pow2 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t} \]

    8. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

    9. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \ell\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{\frac{-1}{3}}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \ell\right) \]

    11. lift-/.f64 29.4

        \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \ell\right) \]

14. Applied rewrites 29.4%

    \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\ell}\right) \]
15. Add Preprocessing

Alternative 20: 20.4% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (* -0.11666666666666667 (* l l)) t))

C

double code(double t, double l, double k) {
	return (-0.11666666666666667 * (l * l)) / t;
}

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 = ((-0.11666666666666667d0) * (l * l)) / t
end function

Java

public static double code(double t, double l, double k) {
	return (-0.11666666666666667 * (l * l)) / t;
}

Python

def code(t, l, k):
	return (-0.11666666666666667 * (l * l)) / t

Julia

function code(t, l, k)
	return Float64(Float64(-0.11666666666666667 * Float64(l * l)) / t)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (-0.11666666666666667 * (l * l)) / t;
end

Wolfram

code[t_, l_, k_] := N[(N[(-0.11666666666666667 * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 28.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 20.4

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 20.4%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{t} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot \left(\ell \cdot \ell\right)}{t} \]

   9. lift-*.f64 20.4

      \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]

10. Applied rewrites 20.4%

    \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]
11. Add Preprocessing

Alternative 21: 20.4% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (/ (* l l) t)))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * ((l * l) / t);
}

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 = (-0.11666666666666667d0) * ((l * l) / t)
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * ((l * l) / t);
}

Python

def code(t, l, k):
	return -0.11666666666666667 * ((l * l) / t)

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(Float64(l * l) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * ((l * l) / t);
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 28.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 20.4

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 20.4%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Add Preprocessing

Alternative 22: 18.2% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

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 = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

def code(t, l, k):
	return -0.11666666666666667 * (l * (l / t))

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(l * Float64(l / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * (l * (l / t));
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 28.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 20.4

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 20.4%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 18.2

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

10. Applied rewrites 18.2%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025086 
(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))))