Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 92.0%

Time: 8.0s

Alternatives: 13

Speedup: 5.8×

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return ((l / sin(k)) * (((l + l) / (k * t)) / sin(k))) * (cos(k) / 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 = ((l / sin(k)) * (((l + l) / (k * t)) / sin(k))) * (cos(k) / k)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 68.1%

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

6. Applied rewrites 72.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. count-2-rev N/A

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

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

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

   15. unpow2 N/A

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

8. Applied rewrites 80.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-commutative N/A

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

   12. count-2-rev N/A

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

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

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

10. Applied rewrites 92.0%

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

Alternative 2: 90.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.000108:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell + \ell}{k \cdot t}\right) \cdot \cos k}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.000108)
   (* (* (/ l (* k k)) (/ l (* k (* k t)))) 2.0)
   (/
    (* (* l (/ (+ l l) (* k t))) (cos k))
    (* (- 0.5 (* (cos (+ k k)) 0.5)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000108) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = ((l * ((l + l) / (k * t))) * cos(k)) / ((0.5 - (cos((k + k)) * 0.5)) * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 0.000108d0) then
        tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0d0
    else
        tmp = ((l * ((l + l) / (k * t))) * cos(k)) / ((0.5d0 - (cos((k + k)) * 0.5d0)) * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000108) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = ((l * ((l + l) / (k * t))) * Math.cos(k)) / ((0.5 - (Math.cos((k + k)) * 0.5)) * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 0.000108:
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0
	else:
		tmp = ((l * ((l + l) / (k * t))) * math.cos(k)) / ((0.5 - (math.cos((k + k)) * 0.5)) * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.000108)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t)))) * 2.0);
	else
		tmp = Float64(Float64(Float64(l * Float64(Float64(l + l) / Float64(k * t))) * cos(k)) / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.000108)
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	else
		tmp = ((l * ((l + l) / (k * t))) * cos(k)) / ((0.5 - (cos((k + k)) * 0.5)) * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.000108], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(l * N[(N[(l + l), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.08e-4

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 63.1%

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 1.08e-4 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

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

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

      15. unpow2 N/A

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

   8. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. frac-times N/A

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

   10. Applied rewrites 79.8%

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

Alternative 3: 82.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 68.1%

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

6. Applied rewrites 72.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. count-2-rev N/A

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

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

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

   15. unpow2 N/A

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

8. Applied rewrites 80.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. count-2-rev N/A

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

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

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

   8. unpow2 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 90.6

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

10. Applied rewrites 90.6%

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

Alternative 4: 80.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.000108:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{0.5 - \cos \left(k + k\right) \cdot 0.5} \cdot \frac{\left(\ell + \ell\right) \cdot \cos k}{\left(k \cdot t\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.000108)
   (* (* (/ l (* k k)) (/ l (* k (* k t)))) 2.0)
   (*
    (/ l (- 0.5 (* (cos (+ k k)) 0.5)))
    (/ (* (+ l l) (cos k)) (* (* k t) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000108) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (l / (0.5 - (cos((k + k)) * 0.5))) * (((l + l) * cos(k)) / ((k * t) * k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 0.000108d0) then
        tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0d0
    else
        tmp = (l / (0.5d0 - (cos((k + k)) * 0.5d0))) * (((l + l) * cos(k)) / ((k * t) * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000108) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (l / (0.5 - (Math.cos((k + k)) * 0.5))) * (((l + l) * Math.cos(k)) / ((k * t) * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 0.000108:
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0
	else:
		tmp = (l / (0.5 - (math.cos((k + k)) * 0.5))) * (((l + l) * math.cos(k)) / ((k * t) * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.000108)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t)))) * 2.0);
	else
		tmp = Float64(Float64(l / Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5))) * Float64(Float64(Float64(l + l) * cos(k)) / Float64(Float64(k * t) * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.000108)
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	else
		tmp = (l / (0.5 - (cos((k + k)) * 0.5))) * (((l + l) * cos(k)) / ((k * t) * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.000108], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(l / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l + l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.08e-4

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 63.1%

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 1.08e-4 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

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

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

      15. unpow2 N/A

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

   8. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 78.4%

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

Alternative 5: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 20:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\cos k}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 20.0)
   (* (* (/ l (* k k)) (/ l (* k (* k t)))) 2.0)
   (*
    (/ (* (+ l l) l) (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t) k))
    (/ (cos k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 20.0) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (((l + l) * l) / (((0.5 - (cos((k + k)) * 0.5)) * t) * k)) * (cos(k) / k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 20.0d0) then
        tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0d0
    else
        tmp = (((l + l) * l) / (((0.5d0 - (cos((k + k)) * 0.5d0)) * t) * k)) * (cos(k) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 20.0) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (((l + l) * l) / (((0.5 - (Math.cos((k + k)) * 0.5)) * t) * k)) * (Math.cos(k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 20.0:
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0
	else:
		tmp = (((l + l) * l) / (((0.5 - (math.cos((k + k)) * 0.5)) * t) * k)) * (math.cos(k) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 20.0)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t)))) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t) * k)) * Float64(cos(k) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 20.0)
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	else
		tmp = (((l + l) * l) / (((0.5 - (cos((k + k)) * 0.5)) * t) * k)) * (cos(k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 20.0], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 20

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 63.1%

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 20 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

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

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

      15. unpow2 N/A

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

   8. Applied rewrites 80.5%

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

   10. Applied rewrites 72.2%

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

Alternative 6: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 20:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \left(\ell + \ell\right)\right) \cdot \cos k}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(k \cdot t\right)\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 20.0)
   (* (* (/ l (* k k)) (/ l (* k (* k t)))) 2.0)
   (/
    (* (* l (+ l l)) (cos k))
    (* (* (- 0.5 (* (cos (+ k k)) 0.5)) (* k t)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 20.0) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = ((l * (l + l)) * cos(k)) / (((0.5 - (cos((k + k)) * 0.5)) * (k * t)) * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 20.0d0) then
        tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0d0
    else
        tmp = ((l * (l + l)) * cos(k)) / (((0.5d0 - (cos((k + k)) * 0.5d0)) * (k * t)) * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 20.0) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = ((l * (l + l)) * Math.cos(k)) / (((0.5 - (Math.cos((k + k)) * 0.5)) * (k * t)) * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 20.0:
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0
	else:
		tmp = ((l * (l + l)) * math.cos(k)) / (((0.5 - (math.cos((k + k)) * 0.5)) * (k * t)) * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 20.0)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t)))) * 2.0);
	else
		tmp = Float64(Float64(Float64(l * Float64(l + l)) * cos(k)) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * Float64(k * t)) * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 20.0)
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	else
		tmp = ((l * (l + l)) * cos(k)) / (((0.5 - (cos((k + k)) * 0.5)) * (k * t)) * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 20.0], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 20

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 63.1%

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 20 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

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

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

      15. unpow2 N/A

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

   8. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lift-/.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. frac-times N/A

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

   10. Applied rewrites 70.3%

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

Alternative 7: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot 2}{t \cdot k}\\ \mathbf{if}\;k \leq 1.05 \cdot 10^{+118}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k \cdot k} \cdot t\_1\right) \cdot \frac{\cos k}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{0.5 - \cos \left(k + k\right) \cdot 0.5} \cdot t\_1\right) \cdot \frac{1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* l 2.0) (* t k))))
   (if (<= k 1.05e+118)
     (*
      (* (/ (fma (* (* k k) l) 0.3333333333333333 l) (* k k)) t_1)
      (/ (cos k) k))
     (* (* (/ l (- 0.5 (* (cos (+ k k)) 0.5))) t_1) (/ 1.0 k)))))

C

double code(double t, double l, double k) {
	double t_1 = (l * 2.0) / (t * k);
	double tmp;
	if (k <= 1.05e+118) {
		tmp = ((fma(((k * k) * l), 0.3333333333333333, l) / (k * k)) * t_1) * (cos(k) / k);
	} else {
		tmp = ((l / (0.5 - (cos((k + k)) * 0.5))) * t_1) * (1.0 / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l * 2.0) / Float64(t * k))
	tmp = 0.0
	if (k <= 1.05e+118)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(k * k) * l), 0.3333333333333333, l) / Float64(k * k)) * t_1) * Float64(cos(k) / k));
	else
		tmp = Float64(Float64(Float64(l / Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5))) * t_1) * Float64(1.0 / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l * 2.0), $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.05e+118], N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * l), $MachinePrecision] * 0.3333333333333333 + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e118

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

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

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

      15. unpow2 N/A

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 60.8

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

   11. Applied rewrites 60.8%

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

   if 1.05e118 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

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

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

      15. unpow2 N/A

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 65.7

          \[\leadsto \left(\frac{\ell}{0.5 - \cos \left(k + k\right) \cdot 0.5} \cdot \frac{\ell \cdot 2}{t \cdot k}\right) \cdot \frac{1}{k} \]

   11. Applied rewrites 65.7%

       \[\leadsto \left(\frac{\ell}{0.5 - \cos \left(k + k\right) \cdot 0.5} \cdot \frac{\ell \cdot 2}{t \cdot k}\right) \cdot \frac{1}{\color{blue}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+200}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\cos k}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 3.6e+200)
   (* (* (/ l (* k k)) (/ l (* k (* k t)))) 2.0)
   (* (/ (* (* l l) 2.0) (* (* (- 0.5 0.5) t) k)) (/ (cos k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.6e+200) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (((l * l) * 2.0) / (((0.5 - 0.5) * t) * k)) * (cos(k) / k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.6d+200) then
        tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0d0
    else
        tmp = (((l * l) * 2.0d0) / (((0.5d0 - 0.5d0) * t) * k)) * (cos(k) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.6e+200) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (((l * l) * 2.0) / (((0.5 - 0.5) * t) * k)) * (Math.cos(k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 3.6e+200:
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0
	else:
		tmp = (((l * l) * 2.0) / (((0.5 - 0.5) * t) * k)) * (math.cos(k) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 3.6e+200)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t)))) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(Float64(0.5 - 0.5) * t) * k)) * Float64(cos(k) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 3.6e+200)
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	else
		tmp = (((l * l) * 2.0) / (((0.5 - 0.5) * t) * k)) * (cos(k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 3.6e+200], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.5999999999999998e200

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 63.1%

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 3.5999999999999998e200 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 36.5%

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

Alternative 9: 73.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{+200}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 4.9e+200)
   (* (* (/ l (* k k)) (/ l (* k (* k t)))) 2.0)
   (/ (* 2.0 (* (cos k) (* l l))) (* (* (- 0.5 0.5) t) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.9e+200) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (2.0 * (cos(k) * (l * l))) / (((0.5 - 0.5) * t) * (k * k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4.9d+200) then
        tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0d0
    else
        tmp = (2.0d0 * (cos(k) * (l * l))) / (((0.5d0 - 0.5d0) * t) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.9e+200) {
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	} else {
		tmp = (2.0 * (Math.cos(k) * (l * l))) / (((0.5 - 0.5) * t) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 4.9e+200:
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0
	else:
		tmp = (2.0 * (math.cos(k) * (l * l))) / (((0.5 - 0.5) * t) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 4.9e+200)
		tmp = Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t)))) * 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k) * Float64(l * l))) / Float64(Float64(Float64(0.5 - 0.5) * t) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 4.9e+200)
		tmp = ((l / (k * k)) * (l / (k * (k * t)))) * 2.0;
	else
		tmp = (2.0 * (cos(k) * (l * l))) / (((0.5 - 0.5) * t) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 4.9e+200], N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.89999999999999982e200

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.1

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

   4. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 63.1%

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 4.89999999999999982e200 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 35.6%

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

Alternative 10: 71.1% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 63.1

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

4. Applied rewrites 63.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-/l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. unpow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

6. Applied rewrites 63.1%

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

8. Applied rewrites 70.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 73.6

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

10. Applied rewrites 73.6%

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

Alternative 11: 69.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 63.1

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

4. Applied rewrites 63.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-/l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. unpow-prod-down N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   11. pow-prod-up N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{\left(2 + 2\right)} \cdot t} \]

   12. metadata-eval N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

6. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t} \cdot 2} \]
7. Step-by-step derivation

8. Applied rewrites 70.3%

   \[\leadsto \color{blue}{\left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2 \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2 \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2 \]

   5. pow2 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left({k}^{2} \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2 \]

   6. associate-*l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \cdot 2 \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \cdot 2 \]

   8. pow2 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \cdot 2 \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \cdot 2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \cdot 2 \]

   11. lift-*.f64 71.1

       \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \cdot 2 \]

10. Applied rewrites 71.1%

    \[\leadsto \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right) \cdot 2 \]
11. Add Preprocessing

Alternative 12: 69.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \cdot \ell\right) \cdot 2 \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* (/ l (* (* (* k k) (* k k)) t)) l) 2.0))

C

double code(double t, double l, double k) {
	return ((l / (((k * k) * (k * k)) * t)) * l) * 2.0;
}

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 = ((l / (((k * k) * (k * k)) * t)) * l) * 2.0d0
end function

Java

public static double code(double t, double l, double k) {
	return ((l / (((k * k) * (k * k)) * t)) * l) * 2.0;
}

Python

def code(t, l, k):
	return ((l / (((k * k) * (k * k)) * t)) * l) * 2.0

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / Float64(Float64(Float64(k * k) * Float64(k * k)) * t)) * l) * 2.0)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / (((k * k) * (k * k)) * t)) * l) * 2.0;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l / N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 \cdot 2\right)} \cdot t} \]

   8. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 63.1

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   5. associate-/l* N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   6. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   8. pow2 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   9. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   10. unpow-prod-down N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   11. pow-prod-up N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{\left(2 + 2\right)} \cdot t} \]

   12. metadata-eval N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

6. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t} \cdot 2} \]
7. Step-by-step derivation

8. Applied rewrites 70.3%

   \[\leadsto \color{blue}{\left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2} \]
9. Step-by-step derivation

10. Applied rewrites 69.1%

    \[\leadsto \left(\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \cdot \ell\right) \cdot \color{blue}{2} \]
11. Add Preprocessing

Alternative 13: 63.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* (+ l l) l) (* (* (* k k) (* k k)) t)))

C

double code(double t, double l, double k) {
	return ((l + l) * l) / (((k * k) * (k * k)) * t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l + l) * l) / (((k * k) * (k * k)) * t)
end function

Java

public static double code(double t, double l, double k) {
	return ((l + l) * l) / (((k * k) * (k * k)) * t);
}

Python

def code(t, l, k):
	return ((l + l) * l) / (((k * k) * (k * k)) * t)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l + l) * l) / Float64(Float64(Float64(k * k) * Float64(k * k)) * t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l + l) * l) / (((k * k) * (k * k)) * t);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 \cdot 2\right)} \cdot t} \]

   8. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 63.1

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   5. associate-/l* N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   6. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   8. pow2 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   9. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   10. unpow-prod-down N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   11. pow-prod-up N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{\left(2 + 2\right)} \cdot t} \]

   12. metadata-eval N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

6. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t} \cdot 2} \]
7. Step-by-step derivation

8. Applied rewrites 70.3%

   \[\leadsto \color{blue}{\left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot 2} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right) \cdot \color{blue}{2} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}}\right) \]

   5. associate-*r/ N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \]

   6. pow2 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot k\right)} \cdot \left(k \cdot t\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \left(k \cdot \color{blue}{t}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   9. associate-*r* N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{t}} \]

   10. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right) \cdot t} \]

   12. pow3 N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\left({k}^{3} \cdot k\right) \cdot t} \]

   13. pow-plus N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{\left(3 + 1\right)} \cdot t} \]

   14. metadata-eval N/A

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   15. associate-/l* N/A

       \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

10. Applied rewrites 63.1%

    \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025135 
(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))))