Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.8% → 96.2%

Time: 8.6s

Alternatives: 14

Speedup: 9.6×

• 38.8% 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: 34.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 74.6

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

5. Applied rewrites 74.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. pow2 N/A

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

   9. frac-times N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. times-frac N/A

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

   16. lower-*.f64 N/A

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

7. Applied rewrites 90.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-cos.f64 96.2

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

9. Applied rewrites 96.2%

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

Alternative 2: 93.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos k \cdot \ell\\ t_2 := {\sin k}^{2} \cdot t\\ \mathbf{if}\;t \leq 2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_2 \cdot k}{\ell} \cdot k}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_2}{t\_1}\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cos k) l)) (t_2 (* (pow (sin k) 2.0) t)))
   (if (<= t 2.6e-53)
     (/ 2.0 (/ (* (/ (* t_2 k) l) k) t_1))
     (/ 2.0 (* (* k (/ t_2 t_1)) (/ k l))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(k) * l;
	double t_2 = pow(sin(k), 2.0) * t;
	double tmp;
	if (t <= 2.6e-53) {
		tmp = 2.0 / ((((t_2 * k) / l) * k) / t_1);
	} else {
		tmp = 2.0 / ((k * (t_2 / t_1)) * (k / l));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(k) * l
    t_2 = (sin(k) ** 2.0d0) * t
    if (t <= 2.6d-53) then
        tmp = 2.0d0 / ((((t_2 * k) / l) * k) / t_1)
    else
        tmp = 2.0d0 / ((k * (t_2 / t_1)) * (k / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(k) * l;
	double t_2 = Math.pow(Math.sin(k), 2.0) * t;
	double tmp;
	if (t <= 2.6e-53) {
		tmp = 2.0 / ((((t_2 * k) / l) * k) / t_1);
	} else {
		tmp = 2.0 / ((k * (t_2 / t_1)) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(k) * l
	t_2 = math.pow(math.sin(k), 2.0) * t
	tmp = 0
	if t <= 2.6e-53:
		tmp = 2.0 / ((((t_2 * k) / l) * k) / t_1)
	else:
		tmp = 2.0 / ((k * (t_2 / t_1)) * (k / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(k) * l)
	t_2 = Float64((sin(k) ^ 2.0) * t)
	tmp = 0.0
	if (t <= 2.6e-53)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_2 * k) / l) * k) / t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_2 / t_1)) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(k) * l;
	t_2 = (sin(k) ^ 2.0) * t;
	tmp = 0.0;
	if (t <= 2.6e-53)
		tmp = 2.0 / ((((t_2 * k) / l) * k) / t_1);
	else
		tmp = 2.0 / ((k * (t_2 / t_1)) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, 2.6e-53], N[(2.0 / N[(N[(N[(N[(t$95$2 * k), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.59999999999999996e-53

   1. Initial program 35.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 95.9

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

   9. Applied rewrites 95.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-/.f64 N/A

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

       10. associate-*r/ N/A

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

       11. *-commutative N/A

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

       12. times-frac N/A

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

       13. associate-/r* N/A

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

       14. lift-/.f64 N/A

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

       15. frac-times N/A

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

       16. *-commutative N/A

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

   11. Applied rewrites 91.4%

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

   if 2.59999999999999996e-53 < t

   1. Initial program 33.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 96.9

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

   9. Applied rewrites 96.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-cos.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*r/ N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lift-sin.f64 N/A

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

       18. lift-pow.f64 N/A

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

       19. *-commutative N/A

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

       20. lower-*.f64 N/A

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

       21. lift-cos.f64 97.0

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

   11. Applied rewrites 97.0%

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

Alternative 3: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4400000000000:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4400000000000.0)
   (/ 2.0 (* (* k (/ (* (pow (sin k) 2.0) t) (* (cos k) l))) (/ k l)))
   (/
    2.0
    (*
     (* (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) (/ t (cos k))) (/ k l))
     (/ k l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4400000000000.0) {
		tmp = 2.0 / ((k * ((pow(sin(k), 2.0) * t) / (cos(k) * l))) * (k / l));
	} else {
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) * (t / cos(k))) * (k / l)) * (k / l));
	}
	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 <= 4400000000000.0d0) then
        tmp = 2.0d0 / ((k * (((sin(k) ** 2.0d0) * t) / (cos(k) * l))) * (k / l))
    else
        tmp = 2.0d0 / ((((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * (t / cos(k))) * (k / l)) * (k / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4400000000000.0) {
		tmp = 2.0 / ((k * ((Math.pow(Math.sin(k), 2.0) * t) / (Math.cos(k) * l))) * (k / l));
	} else {
		tmp = 2.0 / ((((0.5 - (0.5 * Math.cos((2.0 * k)))) * (t / Math.cos(k))) * (k / l)) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4400000000000.0:
		tmp = 2.0 / ((k * ((math.pow(math.sin(k), 2.0) * t) / (math.cos(k) * l))) * (k / l))
	else:
		tmp = 2.0 / ((((0.5 - (0.5 * math.cos((2.0 * k)))) * (t / math.cos(k))) * (k / l)) * (k / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4400000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64((sin(k) ^ 2.0) * t) / Float64(cos(k) * l))) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * Float64(t / cos(k))) * Float64(k / l)) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4400000000000.0)
		tmp = 2.0 / ((k * (((sin(k) ^ 2.0) * t) / (cos(k) * l))) * (k / l));
	else
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) * (t / cos(k))) * (k / l)) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4400000000000.0], N[(2.0 / N[(N[(k * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4e12

   1. Initial program 36.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 95.3

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

   9. Applied rewrites 95.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-cos.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*r/ N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lift-sin.f64 N/A

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

       18. lift-pow.f64 N/A

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

       19. *-commutative N/A

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

       20. lower-*.f64 N/A

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

       21. lift-cos.f64 93.6

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

   11. Applied rewrites 93.6%

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

   if 4.4e12 < k

   1. Initial program 30.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 99.0

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

   9. Applied rewrites 99.0%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 98.6

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

   11. Applied rewrites 98.6%

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

Alternative 4: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 60000:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(t \cdot 0.16666666666666666, k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k}^{2} \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 60000.0)
   (/
    2.0
    (*
     (* (* (fma (* t 0.16666666666666666) (* k k) t) (* k k)) (/ k l))
     (/ k l)))
   (if (<= k 1.1e+122)
     (/
      2.0
      (*
       (/ (* (* k k) t) (cos k))
       (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) (* l l))))
     (/ 2.0 (* (* (* (pow (sin k) 2.0) t) (/ k l)) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 60000.0) {
		tmp = 2.0 / (((fma((t * 0.16666666666666666), (k * k), t) * (k * k)) * (k / l)) * (k / l));
	} else if (k <= 1.1e+122) {
		tmp = 2.0 / ((((k * k) * t) / cos(k)) * ((0.5 - (0.5 * cos((2.0 * k)))) / (l * l)));
	} else {
		tmp = 2.0 / (((pow(sin(k), 2.0) * t) * (k / l)) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 60000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(t * 0.16666666666666666), Float64(k * k), t) * Float64(k * k)) * Float64(k / l)) * Float64(k / l)));
	elseif (k <= 1.1e+122)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) / cos(k)) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / Float64(l * l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) * t) * Float64(k / l)) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 60000.0], N[(2.0 / N[(N[(N[(N[(N[(t * 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.1e+122], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6e4

   1. Initial program 36.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 95.2

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

   9. Applied rewrites 95.2%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. distribute-rgt-out-- N/A

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

       7. metadata-eval N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-*.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 79.8

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

   12. Applied rewrites 79.8%

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

   if 6e4 < k < 1.1e122

   1. Initial program 20.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 1.1e122 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 65.5%

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

Alternative 5: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(t \cdot 0.16666666666666666, k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.0025)
   (/
    2.0
    (*
     (* (* (fma (* t 0.16666666666666666) (* k k) t) (* k k)) (/ k l))
     (/ k l)))
   (/
    2.0
    (*
     (* (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) (/ t (cos k))) (/ k l))
     (/ k l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / (((fma((t * 0.16666666666666666), (k * k), t) * (k * k)) * (k / l)) * (k / l));
	} else {
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) * (t / cos(k))) * (k / l)) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.0025)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(t * 0.16666666666666666), Float64(k * k), t) * Float64(k * k)) * Float64(k / l)) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * Float64(t / cos(k))) * Float64(k / l)) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.0025], N[(2.0 / N[(N[(N[(N[(N[(t * 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 36.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 95.2

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

   9. Applied rewrites 95.2%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. distribute-rgt-out-- N/A

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

       7. metadata-eval N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-*.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 80.0

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

   12. Applied rewrites 80.0%

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

   if 0.00250000000000000005 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 98.9

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

   9. Applied rewrites 98.9%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 98.6

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

   11. Applied rewrites 98.6%

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

Alternative 6: 75.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 74.6

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

5. Applied rewrites 74.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. pow2 N/A

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

   9. frac-times N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. times-frac N/A

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

   16. lower-*.f64 N/A

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

7. Applied rewrites 90.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-cos.f64 96.2

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

9. Applied rewrites 96.2%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 75.6%

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

Alternative 7: 73.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(t \cdot 0.16666666666666666, k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 8.2e+203)
   (/
    2.0
    (*
     (* (* (fma (* t 0.16666666666666666) (* k k) t) (* k k)) (/ k l))
     (/ k l)))
   (/ 2.0 (* (/ (* (* k k) t) (cos k)) (/ (* k k) (* l l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 8.2e+203) {
		tmp = 2.0 / (((fma((t * 0.16666666666666666), (k * k), t) * (k * k)) * (k / l)) * (k / l));
	} else {
		tmp = 2.0 / ((((k * k) * t) / cos(k)) * ((k * k) / (l * l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 8.2e+203)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(t * 0.16666666666666666), Float64(k * k), t) * Float64(k * k)) * Float64(k / l)) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) / cos(k)) * Float64(Float64(k * k) / Float64(l * l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 8.2e+203], N[(2.0 / N[(N[(N[(N[(N[(t * 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.20000000000000033e203

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 90.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 95.9

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

   9. Applied rewrites 95.9%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. distribute-rgt-out-- N/A

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

       7. metadata-eval N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lower-*.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 74.1

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

   12. Applied rewrites 74.1%

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

   if 8.20000000000000033e203 < l

   1. Initial program 35.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

Alternative 8: 73.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 74.6

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

5. Applied rewrites 74.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. pow2 N/A

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

   9. frac-times N/A

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

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

   14. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

   15. times-frac N/A

       \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

7. Applied rewrites 90.6%

   \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \left(\color{blue}{\frac{k}{\ell}} \cdot \frac{k}{\ell}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \left(\frac{\color{blue}{k}}{\ell} \cdot \frac{k}{\ell}\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   5. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   6. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \left(\frac{k}{\color{blue}{\ell}} \cdot \frac{k}{\ell}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{k}{\ell}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{k}{\ell}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{k}}{\ell}} \]

   11. associate-/l* N/A

       \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

   13. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

   16. lift-cos.f64 96.2

       \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

9. Applied rewrites 96.2%

   \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{k}{\ell}}} \]

10. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\left(\left({k}^{2} \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\left({k}^{2} \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

    2. pow2 N/A

       \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

    3. lower-*.f64 73.1

       \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]

12. Applied rewrites 73.1%

    \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}} \]
13. Add Preprocessing

Alternative 9: 71.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (* k k) t) (* (/ k l) (/ k l)))))

C

double code(double t, double l, double k) {
	return 2.0 / (((k * k) * t) * ((k / l) * (k / l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((k * k) * t) * ((k / l) * (k / l)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((k * k) * t) * ((k / l) * (k / l)));
}

Python

def code(t, l, k):
	return 2.0 / (((k * k) * t) * ((k / l) * (k / l)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(k * k) * t) * Float64(Float64(k / l) * Float64(k / l))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k * k) * t) * ((k / l) * (k / l)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 74.6

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 74.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   8. pow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

   9. frac-times N/A

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

   14. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

   15. times-frac N/A

       \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

7. Applied rewrites 90.6%

   \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{\frac{k}{\ell}} \cdot \frac{k}{\ell}\right)} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\color{blue}{\ell}} \cdot \frac{k}{\ell}\right)} \]

   3. lift-*.f64 71.7

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

10. Applied rewrites 71.7%

    \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{\frac{k}{\ell}} \cdot \frac{k}{\ell}\right)} \]
11. Add Preprocessing

Alternative 10: 65.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* k (* k t)) (/ (* k k) (* l l)))))

C

double code(double t, double l, double k) {
	return 2.0 / ((k * (k * t)) * ((k * k) / (l * l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / ((k * (k * t)) * ((k * k) / (l * l)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((k * (k * t)) * ((k * k) / (l * l)));
}

Python

def code(t, l, k):
	return 2.0 / ((k * (k * t)) * ((k * k) / (l * l)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(k * Float64(k * t)) * Float64(Float64(k * k) / Float64(l * l))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((k * (k * t)) * ((k * k) / (l * l)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 74.6

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 74.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]
7. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   3. lift-*.f64 65.9

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

8. Applied rewrites 65.9%

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   5. lower-*.f64 66.3

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

10. Applied rewrites 66.3%

    \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

11. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{{k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]
12. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

    2. lower-*.f64 65.2

       \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

13. Applied rewrites 65.2%

    \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{k \cdot k}{\color{blue}{\ell} \cdot \ell}} \]
14. Add Preprocessing

Alternative 11: 62.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \ell\right)}{t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* (/ 2.0 (* (* k k) (* k k))) (* l l)) t))

C

double code(double t, double l, double k) {
	return ((2.0 / ((k * k) * (k * k))) * (l * l)) / t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((2.0d0 / ((k * k) * (k * k))) * (l * l)) / t
end function

Java

public static double code(double t, double l, double k) {
	return ((2.0 / ((k * k) * (k * k))) * (l * l)) / t;
}

Python

def code(t, l, k):
	return ((2.0 / ((k * k) * (k * k))) * (l * l)) / t

Julia

function code(t, l, k)
	return Float64(Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(l * l)) / t)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((2.0 / ((k * k) * (k * k))) * (l * l)) / t;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. times-frac N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   7. pow2 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-*.f64 60.9

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

5. Applied rewrites 60.9%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   3. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   6. pow2 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot {\ell}^{2}}{t} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   13. lift-*.f64 62.1

       \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

7. Applied rewrites 62.1%

   \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{{k}^{\left(2 + 2\right)}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{\frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \left(\ell \cdot \ell\right)}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \ell\right)}{t} \]

   8. lift-*.f64 62.1

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \ell\right)}{t} \]

9. Applied rewrites 62.1%

   \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \ell\right)}{t} \]
10. Add Preprocessing

Alternative 12: 60.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* (* k k) (* k k))) (/ (* l l) t)))

C

double code(double t, double l, double k) {
	return (2.0 / ((k * k) * (k * k))) * ((l * l) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 / ((k * k) * (k * k))) * ((l * l) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / ((k * k) * (k * k))) * ((l * l) / t);
}

Python

def code(t, l, k):
	return (2.0 / ((k * k) * (k * k))) * ((l * l) / t)

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(Float64(l * l) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. times-frac N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   7. pow2 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-*.f64 60.9

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

5. Applied rewrites 60.9%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2}{{k}^{\left(2 + 2\right)}} \cdot \frac{\ell \cdot \ell}{t} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-*.f64 60.9

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t} \]

7. Applied rewrites 60.9%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]
8. Add Preprocessing

Alternative 13: 19.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (/ (* l l) t)))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * ((l * l) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * ((l * l) / t)
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * ((l * l) / t);
}

Python

def code(t, l, k):
	return -0.11666666666666667 * ((l * l) / t)

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(Float64(l * l) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * ((l * l) / t);
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 19.9

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 19.9%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Add Preprocessing

Alternative 14: 17.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

def code(t, l, k):
	return -0.11666666666666667 * (l * (l / t))

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(l * Float64(l / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * (l * (l / t));
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 19.9

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 19.9%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 17.7

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

10. Applied rewrites 17.7%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(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))))