Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 96.5%

Time: 8.5s

Alternatives: 15

Speedup: 9.6×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.3% 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: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(t, l, k)
	tmp = ((2.0 / ((sin(k) ^ 2.0) * t)) * (l * (cos(k) / 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[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.8

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

5. Applied rewrites 72.8%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
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 90.2%

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

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

    4. *-commutative N/A

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

    5. associate-/l* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. lift-cos.f64 95.5

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

11. Applied rewrites 95.5%

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

Alternative 2: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 0.32:\\ \;\;\;\;\left(\frac{t\_1}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{t\_1}{k}\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= k 0.32)
     (* (* (/ 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 <= 0.32) {
		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 <= 0.32d0) 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 <= 0.32) {
		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 <= 0.32:
		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 <= 0.32)
		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 <= 0.32)
		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, 0.32], 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 < 0.320000000000000007

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 73.6

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

   5. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 86.2%

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

   if 0.320000000000000007 < k

   1. Initial program 27.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 70.5

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

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 89.7%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   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 99.4%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]
   10. 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 99.4

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

   11. Applied rewrites 99.4%

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

Alternative 3: 73.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{t\_1}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(t\_1 \cdot \frac{\ell}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)))
   (if (<= k 4.9e-5)
     (*
      (* (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k)) (/ t_1 k))
      (/ l k))
     (if (<= k 1.3e+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 <= 4.9e-5) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * (t_1 / k)) * (l / k);
	} else if (k <= 1.3e+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 <= 4.9e-5)
		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 <= 1.3e+154)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(t_1 * Float64(l / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t))));
	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_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k, 4.9e-5], 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, 1.3e+154], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if k < 4.9e-5

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 73.6

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

   5. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 94.2%

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

   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. lift-*.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. lift-*.f64 71.8

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

   12. Applied rewrites 71.8%

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

   if 4.9e-5 < k < 1.29999999999999994e154

   1. Initial program 17.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 83.1

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

   5. Applied rewrites 83.1%

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

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

   7. Applied rewrites 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 91.4

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

   9. Applied rewrites 91.4%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 91.4

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

   11. Applied rewrites 91.4%

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

   if 1.29999999999999994e154 < k

   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 56.2

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

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 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 99.4%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 70.7%

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

Alternative 4: 78.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos k \cdot \ell}{k}\\ \mathbf{if}\;k \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot t\_1\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot t\_1\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* (cos k) l) k)))
   (if (<= k 4.9e-5)
     (* (* (/ (/ (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 <= 4.9e-5) {
		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 <= 4.9e-5)
		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, 4.9e-5], 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 < 4.9e-5

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 73.6

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

   5. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 94.2%

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

   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. lift-*.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. lift-*.f64 71.8

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

   12. Applied rewrites 71.8%

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

   if 4.9e-5 < k

   1. Initial program 27.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 70.5

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

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 89.7%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   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 99.4%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]
   10. 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 99.4

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

   11. Applied rewrites 99.4%

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

Alternative 5: 68.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e+83)
   (*
    (*
     (/ (/ (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 <= 1.5e+83) {
		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 <= 1.5e+83)
		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, 1.5e+83], 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 < 1.5e83

   1. Initial program 32.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 74.9

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

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 89.9%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   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 94.7%

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

   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. lift-*.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. lift-*.f64 70.1

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

   12. Applied rewrites 70.1%

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

   if 1.5e83 < k

   1. Initial program 33.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

      1. lift-*.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 99.4%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 68.6%

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

Alternative 6: 67.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+83}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{{\sin k}^{2} \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e+83)
   (*
    (*
     (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k))
     (/ (* (cos k) l) k))
    (/ l k))
   (* (/ 2.0 (* k k)) (* l (/ l (* (pow (sin k) 2.0) t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+83) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * ((cos(k) * l) / k)) * (l / k);
	} else {
		tmp = (2.0 / (k * k)) * (l * (l / (pow(sin(k), 2.0) * t)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e+83)
		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(2.0 / Float64(k * k)) * Float64(l * Float64(l / Float64((sin(k) ^ 2.0) * t))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.5e+83], 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[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e83

   1. Initial program 32.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 74.9

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

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 89.9%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   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 94.7%

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

   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. lift-*.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. lift-*.f64 70.1

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

   12. Applied rewrites 70.1%

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

   if 1.5e83 < k

   1. Initial program 33.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

      1. lift-*.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 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 70.1

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 63.3%

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

Alternative 7: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+98}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{3} \cdot t} \cdot 2\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.7e+98)
   (*
    (*
     (/ (/ (fma 0.6666666666666666 (* k k) 2.0) t) (* k k))
     (/ (* (cos k) l) k))
    (/ l k))
   (* (* (/ l (* (pow k 3.0) t)) 2.0) (/ l k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.7e+98) {
		tmp = (((fma(0.6666666666666666, (k * k), 2.0) / t) / (k * k)) * ((cos(k) * l) / k)) * (l / k);
	} else {
		tmp = ((l / (pow(k, 3.0) * t)) * 2.0) * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.7e+98)
		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((k ^ 3.0) * t)) * 2.0) * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.7e+98], 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[(N[Power[k, 3.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6999999999999997e98

   1. Initial program 32.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 75.1

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

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 89.6%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   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 94.7%

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

   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. lift-*.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. lift-*.f64 69.6

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

   12. Applied rewrites 69.6%

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

   if 4.6999999999999997e98 < k

   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 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 61.7

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

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 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. 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 99.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 64.9

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

   12. Applied rewrites 64.9%

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

Alternative 8: 66.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\left(\cos k \cdot \ell\right) \cdot \frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k \cdot k\right) \cdot \ell, \ell\right)}{t}}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{3} \cdot t} \cdot 2\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2e+111)
   (*
    (/ 2.0 (* k k))
    (*
     (* (cos k) l)
     (/ (/ (fma 0.3333333333333333 (* (* k k) l) l) t) (* k k))))
   (* (* (/ l (* (pow k 3.0) t)) 2.0) (/ l k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e+111) {
		tmp = (2.0 / (k * k)) * ((cos(k) * l) * ((fma(0.3333333333333333, ((k * k) * l), l) / t) / (k * k)));
	} else {
		tmp = ((l / (pow(k, 3.0) * t)) * 2.0) * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2e+111)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(cos(k) * l) * Float64(Float64(fma(0.3333333333333333, Float64(Float64(k * k) * l), l) / t) / Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(l / Float64((k ^ 3.0) * t)) * 2.0) * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2e+111], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(N[(k * k), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[Power[k, 3.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999991e111

   1. Initial program 32.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

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

   7. Applied rewrites 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 87.8

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

   9. Applied rewrites 87.8%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. div-add-rev N/A

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

       4. lower-/.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 68.4

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

   12. Applied rewrites 68.4%

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

   if 1.99999999999999991e111 < k

   1. Initial program 34.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 62.1

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

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 92.9%

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

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 65.5

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

   12. Applied rewrites 65.5%

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

Alternative 9: 72.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+224}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.25e+224)
   (* (/ 2.0 (* k k)) (* (/ l (* k k)) (/ l t)))
   (* (/ 2.0 (* (* k k) t)) (* (/ l k) (/ l k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.25e+224) {
		tmp = (2.0 / (k * k)) * ((l / (k * k)) * (l / t));
	} else {
		tmp = (2.0 / ((k * k) * t)) * ((l / 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) :: tmp
    if (t <= 1.25d+224) then
        tmp = (2.0d0 / (k * k)) * ((l / (k * k)) * (l / t))
    else
        tmp = (2.0d0 / ((k * k) * t)) * ((l / k) * (l / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.25e+224) {
		tmp = (2.0 / (k * k)) * ((l / (k * k)) * (l / t));
	} else {
		tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.25e+224:
		tmp = (2.0 / (k * k)) * ((l / (k * k)) * (l / t))
	else:
		tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.25e+224)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l / Float64(k * k)) * Float64(l / t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l / k) * Float64(l / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.25e+224)
		tmp = (2.0 / (k * k)) * ((l / (k * k)) * (l / t));
	else
		tmp = (2.0 / ((k * k) * t)) * ((l / k) * (l / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.25e+224], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if t < 1.24999999999999991e224

   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 74.4

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

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 76.0%

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

   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. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 76.5

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

   10. Applied rewrites 76.5%

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

   if 1.24999999999999991e224 < t

   1. Initial program 4.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 55.6

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

   5. Applied rewrites 55.6%

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

   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 60.3

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

   8. Applied rewrites 60.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+224}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.4e-171)
   (/ (* (* (/ l t) l) 2.0) (* (* k k) (* k k)))
   (* (/ 2.0 (* k k)) (/ (* l l) (* (* k k) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.4e-171) {
		tmp = (((l / t) * l) * 2.0) / ((k * k) * (k * k));
	} 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 (t <= 3.4d-171) then
        tmp = (((l / t) * l) * 2.0d0) / ((k * k) * (k * k))
    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 (t <= 3.4e-171) {
		tmp = (((l / t) * l) * 2.0) / ((k * k) * (k * k));
	} else {
		tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.4e-171:
		tmp = (((l / t) * l) * 2.0) / ((k * k) * (k * k))
	else:
		tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.4e-171)
		tmp = Float64(Float64(Float64(Float64(l / t) * l) * 2.0) / Float64(Float64(k * k) * Float64(k * k)));
	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 (t <= 3.4e-171)
		tmp = (((l / t) * l) * 2.0) / ((k * k) * (k * k));
	else
		tmp = (2.0 / (k * k)) * ((l * l) / ((k * k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.4e-171], N[(N[(N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * k), $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 t < 3.39999999999999985e-171

   1. Initial program 37.0%

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

   3. Taylor expanded in 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 23.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{\color{blue}{4}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. lift-*.f64 23.9

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

   7. Applied rewrites 23.9%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 73.9

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

   10. Applied rewrites 73.9%

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

   if 3.39999999999999985e-171 < t

   1. Initial program 25.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 66.7

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

   5. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   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 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 79.6

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

   9. Applied rewrites 79.6%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
   11. 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 59.6

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

   12. Applied rewrites 59.6%

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

Alternative 11: 71.4% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double t, double l, double k) {
	return (2.0 / (k * k)) * ((l / (k * k)) * (l / t));
}

Python

def code(t, l, k):
	return (2.0 / (k * k)) * ((l / (k * k)) * (l / t))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.8

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

5. Applied rewrites 72.8%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
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 74.6%

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

8. 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. pow2 N/A

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

   2. times-frac N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-/.f64 73.2

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

10. Applied rewrites 73.2%

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

11. Final simplification 73.2%

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

Alternative 12: 65.0% 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 32.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.8

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

5. Applied rewrites 72.8%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
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 74.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. lift-*.f64 84.5

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

9. Applied rewrites 84.5%

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

10. Taylor expanded in k around 0

    \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
11. 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 64.8

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

12. Applied rewrites 64.8%

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

Alternative 13: 29.6% accurate, 14.4× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* (* l l) -0.3333333333333333) (* (* k k) t)))

C

double code(double t, double l, double k) {
	return ((l * l) * -0.3333333333333333) / ((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 = ((l * l) * (-0.3333333333333333d0)) / ((k * k) * t)
end function

Java

public static double code(double t, double l, double k) {
	return ((l * l) * -0.3333333333333333) / ((k * k) * t);
}

Python

def code(t, l, k):
	return ((l * l) * -0.3333333333333333) / ((k * k) * t)

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.8

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

5. Applied rewrites 72.8%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
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 90.2%

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

8. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 50.2%

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

11. Taylor expanded in k around inf

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

    1. associate-*r/ N/A

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

    2. lower-/.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 28.9

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

13. Applied rewrites 28.9%

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

Alternative 14: 21.0% 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 32.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
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 25.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

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 20.6

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

8. Applied rewrites 20.6%

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

Alternative 15: 18.6% 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 32.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
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 25.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

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 20.6

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

8. Applied rewrites 20.6%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
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 18.3

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

10. Applied rewrites 18.3%

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

Reproduce

herbie shell --seed 2025054 
(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))))