Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.7% → 96.2%

Time: 9.0s

Alternatives: 15

Speedup: 8.6×

• 39.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.7% 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 35.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
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 73.5

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

5. Applied rewrites 73.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
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 91.0%

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

Alternative 2: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+93}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e+93)
   (/ 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 <= 1.95e+93) {
		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 <= 1.95d+93) 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 <= 1.95e+93) {
		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 <= 1.95e+93:
		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 <= 1.95e+93)
		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 <= 1.95e+93)
		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, 1.95e+93], 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 < 1.9500000000000001e93

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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 90.9%

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]
   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 95.4

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

   9. Applied rewrites 95.4%

      \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{k}{\ell}}} \]
   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 92.8

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

   11. Applied rewrites 92.8%

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

   if 1.9500000000000001e93 < k

   1. Initial program 20.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 57.7

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

   5. Applied rewrites 57.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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 91.6%

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]
   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 99.6

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

   9. Applied rewrites 99.6%

      \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{k}{\ell}}} \]
   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 99.5

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

   11. Applied rewrites 99.5%

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

Alternative 3: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.8e-5)
   (/ 2.0 (* (* (* (* k k) t) (/ 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 <= 9.8e-5) {
		tmp = 2.0 / ((((k * k) * t) * (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 <= 9.8d-5) then
        tmp = 2.0d0 / ((((k * k) * t) * (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 <= 9.8e-5) {
		tmp = 2.0 / ((((k * k) * t) * (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 <= 9.8e-5:
		tmp = 2.0 / ((((k * k) * t) * (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 <= 9.8e-5)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) * 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

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.8e-5)
		tmp = 2.0 / ((((k * k) * t) * (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, 9.8e-5], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $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 < 9.8e-5

   1. Initial program 39.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 76.5

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

   5. Applied rewrites 76.5%

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

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

       1. pow2 N/A

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

       2. lift-*.f64 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. lift-*.f64 78.8

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

   12. Applied rewrites 78.8%

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

   if 9.8e-5 < k

   1. Initial program 22.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 63.3

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

   5. Applied rewrites 63.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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 92.1%

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]
   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 98.4

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

   9. Applied rewrites 98.4%

      \[\leadsto \frac{2}{\left(\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{k}{\ell}}} \]
   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 98.3

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

   11. Applied rewrites 98.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \frac{t}{\cos k}\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(k \cdot k\right) \cdot t\\ \mathbf{if}\;k \leq 0.00029:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_1}{\cos k} \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (* k k) t)))
   (if (<= k 0.00029)
     (/ 2.0 (* (* t_1 (/ k l)) (/ k l)))
     (/ 2.0 (* (/ t_1 (cos k)) (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) (* l l)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k * k) * t;
	double tmp;
	if (k <= 0.00029) {
		tmp = 2.0 / ((t_1 * (k / l)) * (k / l));
	} else {
		tmp = 2.0 / ((t_1 / cos(k)) * ((0.5 - (0.5 * cos((2.0 * k)))) / (l * 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) :: tmp
    t_1 = (k * k) * t
    if (k <= 0.00029d0) then
        tmp = 2.0d0 / ((t_1 * (k / l)) * (k / l))
    else
        tmp = 2.0d0 / ((t_1 / cos(k)) * ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) / (l * l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k * k) * t;
	double tmp;
	if (k <= 0.00029) {
		tmp = 2.0 / ((t_1 * (k / l)) * (k / l));
	} else {
		tmp = 2.0 / ((t_1 / Math.cos(k)) * ((0.5 - (0.5 * Math.cos((2.0 * k)))) / (l * l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k * k) * t
	tmp = 0
	if k <= 0.00029:
		tmp = 2.0 / ((t_1 * (k / l)) * (k / l))
	else:
		tmp = 2.0 / ((t_1 / math.cos(k)) * ((0.5 - (0.5 * math.cos((2.0 * k)))) / (l * l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k * k) * t)
	tmp = 0.0
	if (k <= 0.00029)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(k / l)) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 / cos(k)) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / Float64(l * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k * k) * t;
	tmp = 0.0;
	if (k <= 0.00029)
		tmp = 2.0 / ((t_1 * (k / l)) * (k / l));
	else
		tmp = 2.0 / ((t_1 / cos(k)) * ((0.5 - (0.5 * cos((2.0 * k)))) / (l * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k, 0.00029], N[(2.0 / N[(N[(t$95$1 * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 / 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]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9e-4

   1. Initial program 39.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 76.5

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

   5. Applied rewrites 76.5%

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

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

       1. pow2 N/A

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

       2. lift-*.f64 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. lift-*.f64 78.8

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

   12. Applied rewrites 78.8%

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

   if 2.9e-4 < k

   1. Initial program 22.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 63.3

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

   5. Applied rewrites 63.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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 63.4

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

   7. Applied rewrites 63.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00029:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% 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 35.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
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 73.5

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

5. Applied rewrites 73.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
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 91.0%

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

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

12. Applied rewrites 74.1%

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

Alternative 6: 73.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
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 73.5

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

5. Applied rewrites 73.5%

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

6. Taylor expanded in k around 0

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

   1. pow2 N/A

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

   2. pow2 N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 73.1

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

8. Applied rewrites 73.1%

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

Alternative 7: 74.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3200:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3200.0)
   (/ 2.0 (* (* (* (* k k) t) (/ k l)) (/ k l)))
   (/ (* (* (/ l t) l) -0.3333333333333333) (* k k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * (k / l));
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3200.0d0) then
        tmp = 2.0d0 / ((((k * k) * t) * (k / l)) * (k / l))
    else
        tmp = (((l / t) * l) * (-0.3333333333333333d0)) / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * (k / l));
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3200.0:
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * (k / l))
	else:
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3200.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) * Float64(k / l)) * Float64(k / l)));
	else
		tmp = Float64(Float64(Float64(Float64(l / t) * l) * -0.3333333333333333) / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3200.0)
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * (k / l));
	else
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3200.0], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3200

   1. Initial program 39.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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 76.7

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

   5. Applied rewrites 76.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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 90.7%

      \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]
   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 95.3

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

   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. 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. pow2 N/A

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

       2. lift-*.f64 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. lift-*.f64 79.0

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

   12. Applied rewrites 79.0%

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

   if 3200 < k

   1. Initial program 21.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 49.2%

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

   12. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 51.6

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

   14. Applied rewrites 51.6%

       \[\leadsto \frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3200:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 10^{-18}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot 2\right) \cdot \frac{\ell}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1e-18)
   (/ (* (* (/ l t) 2.0) (/ l (* k k))) (* k k))
   (/ 2.0 (* (* (* k k) t) (* (/ k l) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1e-18) {
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k);
	} else {
		tmp = 2.0 / (((k * k) * t) * ((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 (t <= 1d-18) then
        tmp = (((l / t) * 2.0d0) * (l / (k * k))) / (k * k)
    else
        tmp = 2.0d0 / (((k * k) * t) * ((k / l) * (k / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1e-18) {
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k);
	} else {
		tmp = 2.0 / (((k * k) * t) * ((k / l) * (k / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1e-18:
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k)
	else:
		tmp = 2.0 / (((k * k) * t) * ((k / l) * (k / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1e-18)
		tmp = Float64(Float64(Float64(Float64(l / t) * 2.0) * Float64(l / Float64(k * k))) / Float64(k * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t) * Float64(Float64(k / l) * Float64(k / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1e-18)
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k);
	else
		tmp = 2.0 / (((k * k) * t) * ((k / l) * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1e-18], N[(N[(N[(N[(l / t), $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if t < 1.0000000000000001e-18

   1. Initial program 37.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 52.8%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 51.7%

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

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. associate-*l* N/A

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

       5. frac-times N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r/ N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-/.f64 N/A

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

       14. lift-/.f64 N/A

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

       15. lift-*.f64 70.3

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

   14. Applied rewrites 70.3%

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

   if 1.0000000000000001e-18 < t

   1. Initial program 29.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 77.8

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

   5. Applied rewrites 77.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   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 95.4%

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

   8. 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. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 78.1

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

   10. Applied rewrites 78.1%

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

Alternative 9: 72.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3200:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot 2\right) \cdot \frac{\ell}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3200.0)
   (/ (* (* (/ l t) 2.0) (/ l (* k k))) (* k k))
   (/ (* (* (/ l t) l) -0.3333333333333333) (* k k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k);
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3200.0d0) then
        tmp = (((l / t) * 2.0d0) * (l / (k * k))) / (k * k)
    else
        tmp = (((l / t) * l) * (-0.3333333333333333d0)) / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k);
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3200.0:
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k)
	else:
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3200.0)
		tmp = Float64(Float64(Float64(Float64(l / t) * 2.0) * Float64(l / Float64(k * k))) / Float64(k * k));
	else
		tmp = Float64(Float64(Float64(Float64(l / t) * l) * -0.3333333333333333) / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3200.0)
		tmp = (((l / t) * 2.0) * (l / (k * k))) / (k * k);
	else
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3200.0], N[(N[(N[(N[(l / t), $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3200

   1. Initial program 39.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 58.0%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 54.8%

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

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. associate-*l* N/A

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

       5. frac-times N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. associate-*r/ N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-/.f64 N/A

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

       14. lift-/.f64 N/A

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

       15. lift-*.f64 76.0

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

   14. Applied rewrites 76.0%

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

   if 3200 < k

   1. Initial program 21.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 49.2%

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

   12. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 51.6

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

   14. Applied rewrites 51.6%

       \[\leadsto \frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3200:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3200.0)
   (/ 2.0 (* (* (* k k) t) (/ (* k k) (* l l))))
   (/ (* (* (/ l t) l) -0.3333333333333333) (* k k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)));
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3200.0d0) then
        tmp = 2.0d0 / (((k * k) * t) * ((k * k) / (l * l)))
    else
        tmp = (((l / t) * l) * (-0.3333333333333333d0)) / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)));
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3200.0:
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)))
	else:
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3200.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t) * Float64(Float64(k * k) / Float64(l * l))));
	else
		tmp = Float64(Float64(Float64(Float64(l / t) * l) * -0.3333333333333333) / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3200.0)
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)));
	else
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3200.0], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3200

   1. Initial program 39.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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 76.7

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

   5. Applied rewrites 76.7%

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

   6. 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 70.8

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 70.4

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

   11. Applied rewrites 70.4%

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

   if 3200 < k

   1. Initial program 21.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 49.2%

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

   12. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 51.6

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

   14. Applied rewrites 51.6%

       \[\leadsto \frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3200:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3200.0)
   (* (/ 2.0 (* (* k k) (* k k))) (/ (* l l) t))
   (/ (* (* (/ l t) l) -0.3333333333333333) (* k k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t);
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3200.0d0) then
        tmp = (2.0d0 / ((k * k) * (k * k))) * ((l * l) / t)
    else
        tmp = (((l / t) * l) * (-0.3333333333333333d0)) / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3200.0) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t);
	} else {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3200.0:
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t)
	else:
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3200.0)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(Float64(l * l) / t));
	else
		tmp = Float64(Float64(Float64(Float64(l / t) * l) * -0.3333333333333333) / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3200.0)
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t);
	else
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3200.0], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3200

   1. Initial program 39.2%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 65.0

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

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   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 65.0

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

   7. Applied rewrites 65.0%

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

   if 3200 < k

   1. Initial program 21.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 49.2%

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

   12. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 51.6

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

   14. Applied rewrites 51.6%

       \[\leadsto \frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 29.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 6.5e+222)
   (/ (* (* (/ l t) l) -0.3333333333333333) (* k k))
   (* -0.11666666666666667 (* l (/ l t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.5e+222) {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	} else {
		tmp = -0.11666666666666667 * (l * (l / t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 6.5d+222) then
        tmp = (((l / t) * l) * (-0.3333333333333333d0)) / (k * k)
    else
        tmp = (-0.11666666666666667d0) * (l * (l / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.5e+222) {
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	} else {
		tmp = -0.11666666666666667 * (l * (l / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 6.5e+222:
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k)
	else:
		tmp = -0.11666666666666667 * (l * (l / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 6.5e+222)
		tmp = Float64(Float64(Float64(Float64(l / t) * l) * -0.3333333333333333) / Float64(k * k));
	else
		tmp = Float64(-0.11666666666666667 * Float64(l * Float64(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 6.5e+222)
		tmp = (((l / t) * l) * -0.3333333333333333) / (k * k);
	else
		tmp = -0.11666666666666667 * (l * (l / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 6.5e+222], N[(N[(N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.5000000000000006e222

   1. Initial program 35.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 55.8%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 57.2%

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

   12. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 31.4

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

   14. Applied rewrites 31.4%

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

   if 6.5000000000000006e222 < l

   1. Initial program 30.4%

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

   3. Taylor expanded in 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 0.0%

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

   6. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 12.1

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

   8. Applied rewrites 12.1%

      \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   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 12.3

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

   10. Applied rewrites 12.3%

       \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 29.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e+302)
   (/ (* (* l l) -0.3333333333333333) (* (* k k) t))
   (* -0.11666666666666667 (* l (/ l t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+302) {
		tmp = ((l * l) * -0.3333333333333333) / ((k * k) * t);
	} else {
		tmp = -0.11666666666666667 * (l * (l / t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d+302) then
        tmp = ((l * l) * (-0.3333333333333333d0)) / ((k * k) * t)
    else
        tmp = (-0.11666666666666667d0) * (l * (l / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+302) {
		tmp = ((l * l) * -0.3333333333333333) / ((k * k) * t);
	} else {
		tmp = -0.11666666666666667 * (l * (l / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e+302:
		tmp = ((l * l) * -0.3333333333333333) / ((k * k) * t)
	else:
		tmp = -0.11666666666666667 * (l * (l / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+302)
		tmp = Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(Float64(k * k) * t));
	else
		tmp = Float64(-0.11666666666666667 * Float64(l * Float64(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e+302)
		tmp = ((l * l) * -0.3333333333333333) / ((k * k) * t);
	else
		tmp = -0.11666666666666667 * (l * (l / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e+302], N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.0000000000000002e302

   1. Initial program 34.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 73.0%

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

   12. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 39.0

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

   14. Applied rewrites 39.0%

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

   if 2.0000000000000002e302 < (*.f64 l l)

   1. Initial program 37.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 1.4%

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

   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 11.1

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

   8. Applied rewrites 11.1%

      \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   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 11.3

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

   10. Applied rewrites 11.3%

       \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 20.2% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (/ (* l l) t)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * ((l * l) / t)
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * ((l * l) / t);
}

Python

def code(t, l, k):
	return -0.11666666666666667 * ((l * l) / t)

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Taylor expanded in 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 29.5%

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

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 23.2

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

8. Applied rewrites 23.2%

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

Alternative 15: 17.9% accurate, 21.0× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

def code(t, l, k):
	return -0.11666666666666667 * (l * (l / t))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

3. Taylor expanded in 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 29.5%

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

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 23.2

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

8. Applied rewrites 23.2%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
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 22.4

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

10. Applied rewrites 22.4%

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

Reproduce

herbie shell --seed 2025037 
(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))))