Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.4% → 97.7%

Time: 14.5s

Alternatives: 16

Speedup: 3.7×

• 60.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (t l k)
  :precision binary64
  (/
 2
 (*
  (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
  (- (+ 1 (pow (/ k t) 2)) 1))))

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 / N[(N[(N[(N[(N[Power[t, 3], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1 + N[Power[N[(k / t), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 36.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
  :precision binary64
  (/
 2
 (*
  (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
  (- (+ 1 (pow (/ k t) 2)) 1))))

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 / N[(N[(N[(N[(N[Power[t, 3], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1 + N[Power[N[(k / t), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (t l k)
  :precision binary64
  (/ 2 (* (/ k l) (* (/ k l) (* (* (sin k) t) (tan k))))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 73.6%

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

4. Applied rewrites 73.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*r/ N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. tan-quot N/A

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

   15. lift-tan.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 77.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

8. Applied rewrites 91.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

10. Applied rewrites 97.7%

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

Alternative 2: 94.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|\ell\right|}\\ t_2 := \tan k \cdot t\\ \mathbf{if}\;\left|\ell\right| \leq \frac{4763410263543689}{11908525658859223294760121268437066290850060053501019099651935423375594096449911575776314174894302258147533153997065059263030913083222523904}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \sin k\right) \cdot \frac{t\_1 \cdot k}{\left|\ell\right|}}\\ \mathbf{elif}\;\left|\ell\right| \leq 5000000000000000164207812446024630394935062831798058477561567131293735034493939977720006578138637063419747523921612177893242453171057459200:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot k\right) \cdot \left(t\_2 \cdot \frac{k}{\left|\ell\right| \cdot \left|\ell\right|}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ k (fabs l))) (t_2 (* (tan k) t)))
  (if (<=
       (fabs l)
       4763410263543689/11908525658859223294760121268437066290850060053501019099651935423375594096449911575776314174894302258147533153997065059263030913083222523904)
    (/ 2 (* (* t_2 (sin k)) (/ (* t_1 k) (fabs l))))
    (if (<=
         (fabs l)
         5000000000000000164207812446024630394935062831798058477561567131293735034493939977720006578138637063419747523921612177893242453171057459200)
      (/ 2 (* (* (sin k) k) (* t_2 (/ k (* (fabs l) (fabs l))))))
      (/ 2 (* (* t (* (tan k) (sin k))) (* t_1 t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(l);
	double t_2 = tan(k) * t;
	double tmp;
	if (fabs(l) <= 4e-124) {
		tmp = 2.0 / ((t_2 * sin(k)) * ((t_1 * k) / fabs(l)));
	} else if (fabs(l) <= 5e+138) {
		tmp = 2.0 / ((sin(k) * k) * (t_2 * (k / (fabs(l) * fabs(l)))));
	} else {
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (t_1 * t_1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k / abs(l)
    t_2 = tan(k) * t
    if (abs(l) <= 4d-124) then
        tmp = 2.0d0 / ((t_2 * sin(k)) * ((t_1 * k) / abs(l)))
    else if (abs(l) <= 5d+138) then
        tmp = 2.0d0 / ((sin(k) * k) * (t_2 * (k / (abs(l) * abs(l)))))
    else
        tmp = 2.0d0 / ((t * (tan(k) * sin(k))) * (t_1 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k / Math.abs(l);
	double t_2 = Math.tan(k) * t;
	double tmp;
	if (Math.abs(l) <= 4e-124) {
		tmp = 2.0 / ((t_2 * Math.sin(k)) * ((t_1 * k) / Math.abs(l)));
	} else if (Math.abs(l) <= 5e+138) {
		tmp = 2.0 / ((Math.sin(k) * k) * (t_2 * (k / (Math.abs(l) * Math.abs(l)))));
	} else {
		tmp = 2.0 / ((t * (Math.tan(k) * Math.sin(k))) * (t_1 * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k / math.fabs(l)
	t_2 = math.tan(k) * t
	tmp = 0
	if math.fabs(l) <= 4e-124:
		tmp = 2.0 / ((t_2 * math.sin(k)) * ((t_1 * k) / math.fabs(l)))
	elif math.fabs(l) <= 5e+138:
		tmp = 2.0 / ((math.sin(k) * k) * (t_2 * (k / (math.fabs(l) * math.fabs(l)))))
	else:
		tmp = 2.0 / ((t * (math.tan(k) * math.sin(k))) * (t_1 * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / abs(l))
	t_2 = Float64(tan(k) * t)
	tmp = 0.0
	if (abs(l) <= 4e-124)
		tmp = Float64(2.0 / Float64(Float64(t_2 * sin(k)) * Float64(Float64(t_1 * k) / abs(l))));
	elseif (abs(l) <= 5e+138)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * k) * Float64(t_2 * Float64(k / Float64(abs(l) * abs(l))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(tan(k) * sin(k))) * Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k / abs(l);
	t_2 = tan(k) * t;
	tmp = 0.0;
	if (abs(l) <= 4e-124)
		tmp = 2.0 / ((t_2 * sin(k)) * ((t_1 * k) / abs(l)));
	elseif (abs(l) <= 5e+138)
		tmp = 2.0 / ((sin(k) * k) * (t_2 * (k / (abs(l) * abs(l)))));
	else
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (t_1 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 4763410263543689/11908525658859223294760121268437066290850060053501019099651935423375594096449911575776314174894302258147533153997065059263030913083222523904], N[(2 / N[(N[(t$95$2 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * k), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 5000000000000000164207812446024630394935062831798058477561567131293735034493939977720006578138637063419747523921612177893242453171057459200], N[(2 / N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] * N[(t$95$2 * N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(t * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.9999999999999997e-124

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.1%

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

   8. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 90.5%

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

   10. Applied rewrites 90.5%

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

   if 3.9999999999999997e-124 < l < 5.0000000000000002e138

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 78.9%

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

   10. Applied rewrites 78.9%

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

   if 5.0000000000000002e138 < l

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 91.2%

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

   8. Applied rewrites 91.2%

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

Alternative 3: 92.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|\ell\right|}\\ t_2 := \tan k \cdot t\\ \mathbf{if}\;\left|\ell\right| \leq \frac{467650163306275}{570305077202774402330146450712536415611653563406069991322399064737409395901084673313239101414671424972330288696699300913836088090171929749781582664965276773531499334295033118760359274030825472}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_2 \cdot {k}^{2}\right) \cdot t\_1}{\left|\ell\right|}}\\ \mathbf{elif}\;\left|\ell\right| \leq 5000000000000000164207812446024630394935062831798058477561567131293735034493939977720006578138637063419747523921612177893242453171057459200:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \sin k\right) \cdot \frac{k}{\left|\ell\right| \cdot \left|\ell\right|}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ k (fabs l))) (t_2 (* (tan k) t)))
  (if (<=
       (fabs l)
       467650163306275/570305077202774402330146450712536415611653563406069991322399064737409395901084673313239101414671424972330288696699300913836088090171929749781582664965276773531499334295033118760359274030825472)
    (/ 2 (/ (* (* t_2 (pow k 2)) t_1) (fabs l)))
    (if (<=
         (fabs l)
         5000000000000000164207812446024630394935062831798058477561567131293735034493939977720006578138637063419747523921612177893242453171057459200)
      (/ 2 (* (* (* t_2 (sin k)) (/ k (* (fabs l) (fabs l)))) k))
      (/ 2 (* (* t (* (tan k) (sin k))) (* t_1 t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(l);
	double t_2 = tan(k) * t;
	double tmp;
	if (fabs(l) <= 8.2e-178) {
		tmp = 2.0 / (((t_2 * pow(k, 2.0)) * t_1) / fabs(l));
	} else if (fabs(l) <= 5e+138) {
		tmp = 2.0 / (((t_2 * sin(k)) * (k / (fabs(l) * fabs(l)))) * k);
	} else {
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (t_1 * t_1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k / abs(l)
    t_2 = tan(k) * t
    if (abs(l) <= 8.2d-178) then
        tmp = 2.0d0 / (((t_2 * (k ** 2.0d0)) * t_1) / abs(l))
    else if (abs(l) <= 5d+138) then
        tmp = 2.0d0 / (((t_2 * sin(k)) * (k / (abs(l) * abs(l)))) * k)
    else
        tmp = 2.0d0 / ((t * (tan(k) * sin(k))) * (t_1 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k / Math.abs(l);
	double t_2 = Math.tan(k) * t;
	double tmp;
	if (Math.abs(l) <= 8.2e-178) {
		tmp = 2.0 / (((t_2 * Math.pow(k, 2.0)) * t_1) / Math.abs(l));
	} else if (Math.abs(l) <= 5e+138) {
		tmp = 2.0 / (((t_2 * Math.sin(k)) * (k / (Math.abs(l) * Math.abs(l)))) * k);
	} else {
		tmp = 2.0 / ((t * (Math.tan(k) * Math.sin(k))) * (t_1 * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k / math.fabs(l)
	t_2 = math.tan(k) * t
	tmp = 0
	if math.fabs(l) <= 8.2e-178:
		tmp = 2.0 / (((t_2 * math.pow(k, 2.0)) * t_1) / math.fabs(l))
	elif math.fabs(l) <= 5e+138:
		tmp = 2.0 / (((t_2 * math.sin(k)) * (k / (math.fabs(l) * math.fabs(l)))) * k)
	else:
		tmp = 2.0 / ((t * (math.tan(k) * math.sin(k))) * (t_1 * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / abs(l))
	t_2 = Float64(tan(k) * t)
	tmp = 0.0
	if (abs(l) <= 8.2e-178)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * (k ^ 2.0)) * t_1) / abs(l)));
	elseif (abs(l) <= 5e+138)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * sin(k)) * Float64(k / Float64(abs(l) * abs(l)))) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(tan(k) * sin(k))) * Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k / abs(l);
	t_2 = tan(k) * t;
	tmp = 0.0;
	if (abs(l) <= 8.2e-178)
		tmp = 2.0 / (((t_2 * (k ^ 2.0)) * t_1) / abs(l));
	elseif (abs(l) <= 5e+138)
		tmp = 2.0 / (((t_2 * sin(k)) * (k / (abs(l) * abs(l)))) * k);
	else
		tmp = 2.0 / ((t * (tan(k) * sin(k))) * (t_1 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 467650163306275/570305077202774402330146450712536415611653563406069991322399064737409395901084673313239101414671424972330288696699300913836088090171929749781582664965276773531499334295033118760359274030825472], N[(2 / N[(N[(N[(t$95$2 * N[Power[k, 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 5000000000000000164207812446024630394935062831798058477561567131293735034493939977720006578138637063419747523921612177893242453171057459200], N[(2 / N[(N[(N[(t$95$2 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(t * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.1999999999999998e-178

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 72.4%

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

   11. Applied rewrites 72.4%

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

   if 8.1999999999999998e-178 < l < 5.0000000000000002e138

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.9%

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 79.1%

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

   8. Applied rewrites 79.1%

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

   if 5.0000000000000002e138 < l

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 91.2%

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

   8. Applied rewrites 91.2%

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

Alternative 4: 85.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \tan \left(\left|k\right|\right) \cdot t\\ \mathbf{if}\;\left|k\right| \leq \frac{4911261142184431}{37778931862957161709568}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_1 \cdot {\left(\left|k\right|\right)}^{2}\right) \cdot \frac{\left|k\right|}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \sin \left(\left|k\right|\right)\right) \cdot \frac{\left|k\right|}{\ell \cdot \ell}\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (tan (fabs k)) t)))
  (if (<= (fabs k) 4911261142184431/37778931862957161709568)
    (/ 2 (/ (* (* t_1 (pow (fabs k) 2)) (/ (fabs k) l)) l))
    (/
     2
     (* (* (* t_1 (sin (fabs k))) (/ (fabs k) (* l l))) (fabs k))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(fabs(k)) * t;
	double tmp;
	if (fabs(k) <= 1.3e-7) {
		tmp = 2.0 / (((t_1 * pow(fabs(k), 2.0)) * (fabs(k) / l)) / l);
	} else {
		tmp = 2.0 / (((t_1 * sin(fabs(k))) * (fabs(k) / (l * l))) * fabs(k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = tan(abs(k)) * t
    if (abs(k) <= 1.3d-7) then
        tmp = 2.0d0 / (((t_1 * (abs(k) ** 2.0d0)) * (abs(k) / l)) / l)
    else
        tmp = 2.0d0 / (((t_1 * sin(abs(k))) * (abs(k) / (l * l))) * abs(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(Math.abs(k)) * t;
	double tmp;
	if (Math.abs(k) <= 1.3e-7) {
		tmp = 2.0 / (((t_1 * Math.pow(Math.abs(k), 2.0)) * (Math.abs(k) / l)) / l);
	} else {
		tmp = 2.0 / (((t_1 * Math.sin(Math.abs(k))) * (Math.abs(k) / (l * l))) * Math.abs(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(math.fabs(k)) * t
	tmp = 0
	if math.fabs(k) <= 1.3e-7:
		tmp = 2.0 / (((t_1 * math.pow(math.fabs(k), 2.0)) * (math.fabs(k) / l)) / l)
	else:
		tmp = 2.0 / (((t_1 * math.sin(math.fabs(k))) * (math.fabs(k) / (l * l))) * math.fabs(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(abs(k)) * t)
	tmp = 0.0
	if (abs(k) <= 1.3e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * (abs(k) ^ 2.0)) * Float64(abs(k) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * sin(abs(k))) * Float64(abs(k) / Float64(l * l))) * abs(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(abs(k)) * t;
	tmp = 0.0;
	if (abs(k) <= 1.3e-7)
		tmp = 2.0 / (((t_1 * (abs(k) ^ 2.0)) * (abs(k) / l)) / l);
	else
		tmp = 2.0 / (((t_1 * sin(abs(k))) * (abs(k) / (l * l))) * abs(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4911261142184431/37778931862957161709568], N[(2 / N[(N[(N[(t$95$1 * N[Power[N[Abs[k], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(t$95$1 * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-7

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 72.4%

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

   11. Applied rewrites 72.4%

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

   if 1.3e-7 < k

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.9%

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 79.1%

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

   8. Applied rewrites 79.1%

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

Alternative 5: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq \frac{4911261142184431}{37778931862957161709568}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\left|k\right|\right)}^{3} \cdot t\right) \cdot \frac{\left|k\right|}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan \left(\left|k\right|\right) \cdot t\right) \cdot \sin \left(\left|k\right|\right)\right) \cdot \frac{\left|k\right|}{\ell \cdot \ell}\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 4911261142184431/37778931862957161709568)
  (/ 2 (/ (* (* (pow (fabs k) 3) t) (/ (fabs k) l)) l))
  (/
   2
   (*
    (* (* (* (tan (fabs k)) t) (sin (fabs k))) (/ (fabs k) (* l l)))
    (fabs k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.3e-7) {
		tmp = 2.0 / (((pow(fabs(k), 3.0) * t) * (fabs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((((tan(fabs(k)) * t) * sin(fabs(k))) * (fabs(k) / (l * l))) * fabs(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 (abs(k) <= 1.3d-7) then
        tmp = 2.0d0 / ((((abs(k) ** 3.0d0) * t) * (abs(k) / l)) / l)
    else
        tmp = 2.0d0 / ((((tan(abs(k)) * t) * sin(abs(k))) * (abs(k) / (l * l))) * abs(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 1.3e-7) {
		tmp = 2.0 / (((Math.pow(Math.abs(k), 3.0) * t) * (Math.abs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((((Math.tan(Math.abs(k)) * t) * Math.sin(Math.abs(k))) * (Math.abs(k) / (l * l))) * Math.abs(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1.3e-7:
		tmp = 2.0 / (((math.pow(math.fabs(k), 3.0) * t) * (math.fabs(k) / l)) / l)
	else:
		tmp = 2.0 / ((((math.tan(math.fabs(k)) * t) * math.sin(math.fabs(k))) * (math.fabs(k) / (l * l))) * math.fabs(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.3e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64((abs(k) ^ 3.0) * t) * Float64(abs(k) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(abs(k)) * t) * sin(abs(k))) * Float64(abs(k) / Float64(l * l))) * abs(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1.3e-7)
		tmp = 2.0 / ((((abs(k) ^ 3.0) * t) * (abs(k) / l)) / l);
	else
		tmp = 2.0 / ((((tan(abs(k)) * t) * sin(abs(k))) * (abs(k) / (l * l))) * abs(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 4911261142184431/37778931862957161709568], N[(2 / N[(N[(N[(N[Power[N[Abs[k], $MachinePrecision], 3], $MachinePrecision] * t), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-7

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 70.5%

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

   11. Applied rewrites 70.5%

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

   if 1.3e-7 < k

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.9%

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 79.1%

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

   8. Applied rewrites 79.1%

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

Alternative 6: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq \frac{4911261142184431}{37778931862957161709568}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\left|k\right|\right)}^{3} \cdot t\right) \cdot \frac{\left|k\right|}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin \left(\left|k\right|\right) \cdot \left|k\right|\right) \cdot \left(\left(\tan \left(\left|k\right|\right) \cdot t\right) \cdot \frac{\left|k\right|}{\ell \cdot \ell}\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 4911261142184431/37778931862957161709568)
  (/ 2 (/ (* (* (pow (fabs k) 3) t) (/ (fabs k) l)) l))
  (/
   2
   (*
    (* (sin (fabs k)) (fabs k))
    (* (* (tan (fabs k)) t) (/ (fabs k) (* l l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.3e-7) {
		tmp = 2.0 / (((pow(fabs(k), 3.0) * t) * (fabs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((sin(fabs(k)) * fabs(k)) * ((tan(fabs(k)) * t) * (fabs(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) :: tmp
    if (abs(k) <= 1.3d-7) then
        tmp = 2.0d0 / ((((abs(k) ** 3.0d0) * t) * (abs(k) / l)) / l)
    else
        tmp = 2.0d0 / ((sin(abs(k)) * abs(k)) * ((tan(abs(k)) * t) * (abs(k) / (l * l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 1.3e-7) {
		tmp = 2.0 / (((Math.pow(Math.abs(k), 3.0) * t) * (Math.abs(k) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.sin(Math.abs(k)) * Math.abs(k)) * ((Math.tan(Math.abs(k)) * t) * (Math.abs(k) / (l * l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1.3e-7:
		tmp = 2.0 / (((math.pow(math.fabs(k), 3.0) * t) * (math.fabs(k) / l)) / l)
	else:
		tmp = 2.0 / ((math.sin(math.fabs(k)) * math.fabs(k)) * ((math.tan(math.fabs(k)) * t) * (math.fabs(k) / (l * l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.3e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64((abs(k) ^ 3.0) * t) * Float64(abs(k) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(abs(k)) * abs(k)) * Float64(Float64(tan(abs(k)) * t) * Float64(abs(k) / Float64(l * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1.3e-7)
		tmp = 2.0 / ((((abs(k) ^ 3.0) * t) * (abs(k) / l)) / l);
	else
		tmp = 2.0 / ((sin(abs(k)) * abs(k)) * ((tan(abs(k)) * t) * (abs(k) / (l * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 4911261142184431/37778931862957161709568], N[(2 / N[(N[(N[(N[Power[N[Abs[k], $MachinePrecision], 3], $MachinePrecision] * t), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-7

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 70.5%

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

   11. Applied rewrites 70.5%

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

   if 1.3e-7 < k

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 78.9%

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

   10. Applied rewrites 78.9%

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

Alternative 7: 82.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq \frac{4911261142184431}{37778931862957161709568}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\left|k\right|\right)}^{3} \cdot t\right) \cdot \frac{\left|k\right|}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left|k\right| \cdot \left(\left|k\right| \cdot \left(\left(\tan \left(\left|k\right|\right) \cdot \sin \left(\left|k\right|\right)\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 4911261142184431/37778931862957161709568)
  (/ 2 (/ (* (* (pow (fabs k) 3) t) (/ (fabs k) l)) l))
  (/
   2
   (*
    (fabs k)
    (*
     (fabs k)
     (* (* (tan (fabs k)) (sin (fabs k))) (/ t (* l l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.3e-7) {
		tmp = 2.0 / (((pow(fabs(k), 3.0) * t) * (fabs(k) / l)) / l);
	} else {
		tmp = 2.0 / (fabs(k) * (fabs(k) * ((tan(fabs(k)) * sin(fabs(k))) * (t / (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) :: tmp
    if (abs(k) <= 1.3d-7) then
        tmp = 2.0d0 / ((((abs(k) ** 3.0d0) * t) * (abs(k) / l)) / l)
    else
        tmp = 2.0d0 / (abs(k) * (abs(k) * ((tan(abs(k)) * sin(abs(k))) * (t / (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 1.3e-7) {
		tmp = 2.0 / (((Math.pow(Math.abs(k), 3.0) * t) * (Math.abs(k) / l)) / l);
	} else {
		tmp = 2.0 / (Math.abs(k) * (Math.abs(k) * ((Math.tan(Math.abs(k)) * Math.sin(Math.abs(k))) * (t / (l * l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1.3e-7:
		tmp = 2.0 / (((math.pow(math.fabs(k), 3.0) * t) * (math.fabs(k) / l)) / l)
	else:
		tmp = 2.0 / (math.fabs(k) * (math.fabs(k) * ((math.tan(math.fabs(k)) * math.sin(math.fabs(k))) * (t / (l * l)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.3e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64((abs(k) ^ 3.0) * t) * Float64(abs(k) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(abs(k) * Float64(abs(k) * Float64(Float64(tan(abs(k)) * sin(abs(k))) * Float64(t / Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1.3e-7)
		tmp = 2.0 / ((((abs(k) ^ 3.0) * t) * (abs(k) / l)) / l);
	else
		tmp = 2.0 / (abs(k) * (abs(k) * ((tan(abs(k)) * sin(abs(k))) * (t / (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 4911261142184431/37778931862957161709568], N[(2 / N[(N[(N[(N[Power[N[Abs[k], $MachinePrecision], 3], $MachinePrecision] * t), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-7

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 70.5%

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

   11. Applied rewrites 70.5%

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

   if 1.3e-7 < k

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

   6. Applied rewrites 75.1%

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

Alternative 8: 72.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 5600000000000000129321311295108080674007094059169515942411161637667116262204963057648266062460524262220381358454248015500955464633560891638505766413260665039987376016101159633050213350899712:\\ \;\;\;\;\frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\left|\ell\right|}}{\left|\ell\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \frac{k}{\left|\ell\right| \cdot \left|\ell\right|}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{\cos k}}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<=
     (fabs l)
     5600000000000000129321311295108080674007094059169515942411161637667116262204963057648266062460524262220381358454248015500955464633560891638505766413260665039987376016101159633050213350899712)
  (/ 2 (/ (* (* (pow k 3) t) (/ k (fabs l))) (fabs l)))
  (/
   2
   (/
    (* (* k (/ k (* (fabs l) (fabs l)))) (* (- 1/2 1/2) t))
    (cos k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 5.6e+189) {
		tmp = 2.0 / (((pow(k, 3.0) * t) * (k / fabs(l))) / fabs(l));
	} else {
		tmp = 2.0 / (((k * (k / (fabs(l) * fabs(l)))) * ((0.5 - 0.5) * t)) / cos(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 (abs(l) <= 5.6d+189) then
        tmp = 2.0d0 / ((((k ** 3.0d0) * t) * (k / abs(l))) / abs(l))
    else
        tmp = 2.0d0 / (((k * (k / (abs(l) * abs(l)))) * ((0.5d0 - 0.5d0) * t)) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 5.6e+189) {
		tmp = 2.0 / (((Math.pow(k, 3.0) * t) * (k / Math.abs(l))) / Math.abs(l));
	} else {
		tmp = 2.0 / (((k * (k / (Math.abs(l) * Math.abs(l)))) * ((0.5 - 0.5) * t)) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 5.6e+189:
		tmp = 2.0 / (((math.pow(k, 3.0) * t) * (k / math.fabs(l))) / math.fabs(l))
	else:
		tmp = 2.0 / (((k * (k / (math.fabs(l) * math.fabs(l)))) * ((0.5 - 0.5) * t)) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 5.6e+189)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 3.0) * t) * Float64(k / abs(l))) / abs(l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / Float64(abs(l) * abs(l)))) * Float64(Float64(0.5 - 0.5) * t)) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 5.6e+189)
		tmp = 2.0 / ((((k ^ 3.0) * t) * (k / abs(l))) / abs(l));
	else
		tmp = 2.0 / (((k * (k / (abs(l) * abs(l)))) * ((0.5 - 0.5) * t)) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 5600000000000000129321311295108080674007094059169515942411161637667116262204963057648266062460524262220381358454248015500955464633560891638505766413260665039987376016101159633050213350899712], N[(2 / N[(N[(N[(N[Power[k, 3], $MachinePrecision] * t), $MachinePrecision] * N[(k / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(k * N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1/2 - 1/2), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.6000000000000001e189

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 91.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 70.5%

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

   11. Applied rewrites 70.5%

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

   if 5.6000000000000001e189 < l

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 36.4%

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

Alternative 9: 70.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq \frac{1147258064484217}{8498207885068273579033411304839498037273489883632510771191106211206456957773635883826600036243668570702229271779944016245545269402443315830552319660265397631101300333366291504507650048}:\\ \;\;\;\;2 \cdot \left(\left|\ell\right| \cdot \frac{{k}^{-4} \cdot \left|\ell\right|}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(k \cdot \frac{k}{\left|\ell\right| \cdot \left|\ell\right|}\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<=
     (fabs l)
     1147258064484217/8498207885068273579033411304839498037273489883632510771191106211206456957773635883826600036243668570702229271779944016245545269402443315830552319660265397631101300333366291504507650048)
  (* 2 (* (fabs l) (/ (* (pow k -4) (fabs l)) t)))
  (/ 2 (* (* t (pow k 2)) (* k (/ k (* (fabs l) (fabs l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 1.35e-169) {
		tmp = 2.0 * (fabs(l) * ((pow(k, -4.0) * fabs(l)) / t));
	} else {
		tmp = 2.0 / ((t * pow(k, 2.0)) * (k * (k / (fabs(l) * fabs(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 (abs(l) <= 1.35d-169) then
        tmp = 2.0d0 * (abs(l) * (((k ** (-4.0d0)) * abs(l)) / t))
    else
        tmp = 2.0d0 / ((t * (k ** 2.0d0)) * (k * (k / (abs(l) * abs(l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 1.35e-169) {
		tmp = 2.0 * (Math.abs(l) * ((Math.pow(k, -4.0) * Math.abs(l)) / t));
	} else {
		tmp = 2.0 / ((t * Math.pow(k, 2.0)) * (k * (k / (Math.abs(l) * Math.abs(l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 1.35e-169:
		tmp = 2.0 * (math.fabs(l) * ((math.pow(k, -4.0) * math.fabs(l)) / t))
	else:
		tmp = 2.0 / ((t * math.pow(k, 2.0)) * (k * (k / (math.fabs(l) * math.fabs(l)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 1.35e-169)
		tmp = Float64(2.0 * Float64(abs(l) * Float64(Float64((k ^ -4.0) * abs(l)) / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k ^ 2.0)) * Float64(k * Float64(k / Float64(abs(l) * abs(l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 1.35e-169)
		tmp = 2.0 * (abs(l) * (((k ^ -4.0) * abs(l)) / t));
	else
		tmp = 2.0 / ((t * (k ^ 2.0)) * (k * (k / (abs(l) * abs(l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 1147258064484217/8498207885068273579033411304839498037273489883632510771191106211206456957773635883826600036243668570702229271779944016245545269402443315830552319660265397631101300333366291504507650048], N[(2 * N[(N[Abs[l], $MachinePrecision] * N[(N[(N[Power[k, -4], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(t * N[Power[k, 2], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3500000000000001e-169

   1. Initial program 36.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 63.0%

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

   4. Applied rewrites 63.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 69.0%

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

   6. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.7%

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

   8. Applied rewrites 69.7%

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

   if 1.3500000000000001e-169 < l

   1. Initial program 36.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(k \cdot \frac{k}{\ell \cdot \ell}\right)} \]
   8. Step-by-step derivation

      1. lower-pow.f64 65.7%

         \[\leadsto \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(k \cdot \frac{k}{\ell \cdot \ell}\right)} \]

   9. Applied rewrites 65.7%

      \[\leadsto \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(k \cdot \frac{k}{\ell \cdot \ell}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.5% accurate, 3.2× speedup

Math

\[\frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (/ 2 (/ (* (* (pow k 3) t) (/ k l)) l)))

C

double code(double t, double l, double k) {
	return 2.0 / (((pow(k, 3.0) * t) * (k / l)) / l);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / ((((k ** 3.0d0) * t) * (k / l)) / l)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((Math.pow(k, 3.0) * t) * (k / l)) / l);
}

Python

def code(t, l, k):
	return 2.0 / (((math.pow(k, 3.0) * t) * (k / l)) / l)

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64((k ^ 3.0) * t) * Float64(k / l)) / l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((((k ^ 3.0) * t) * (k / l)) / l);
end

Wolfram

code[t_, l_, k_] := N[(2 / N[(N[(N[(N[Power[k, 3], $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

   9. lower-cos.f64 73.6%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

4. Applied rewrites 73.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   4. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{\color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\left(t \cdot \frac{\sin k \cdot \sin k}{\cos k}\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\sin k \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\sin k \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   13. lift-cos.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\sin k \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   14. tan-quot N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   15. lift-tan.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{\color{blue}{{k}^{2}}}{{\ell}^{2}}} \]

   17. *-commutative N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{{k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

6. Applied rewrites 77.4%

   \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell \cdot \color{blue}{\ell}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\ell}}{\color{blue}{\ell}}} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

8. Applied rewrites 91.2%

   \[\leadsto \frac{2}{\frac{\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \frac{k}{\ell}}{\color{blue}{\ell}}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]

    2. lower-pow.f64 70.5%

       \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]

11. Applied rewrites 70.5%

    \[\leadsto \frac{2}{\frac{\left({k}^{3} \cdot t\right) \cdot \frac{k}{\ell}}{\ell}} \]
12. Add Preprocessing

Alternative 11: 69.7% accurate, 3.6× speedup

Math

\[\left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (* l (/ l (* (pow k 4) t))) 2))

C

double code(double t, double l, double k) {
	return (l * (l / (pow(k, 4.0) * t))) * 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * (l / ((k ** 4.0d0) * t))) * 2.0d0
end function

Java

public static double code(double t, double l, double k) {
	return (l * (l / (Math.pow(k, 4.0) * t))) * 2.0;
}

Python

def code(t, l, k):
	return (l * (l / (math.pow(k, 4.0) * t))) * 2.0

Julia

function code(t, l, k)
	return Float64(Float64(l * Float64(l / Float64((k ^ 4.0) * t))) * 2.0)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * (l / ((k ^ 4.0) * t))) * 2.0;
end

Wolfram

code[t_, l_, k_] := N[(N[(l * N[(l / N[(N[Power[k, 4], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 63.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 63.0%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 63.0%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 69.2%

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 69.2%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Add Preprocessing

Alternative 12: 69.2% accurate, 3.6× speedup

Math

\[2 \cdot \left(\ell \cdot \frac{{k}^{-4} \cdot \ell}{t}\right) \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* 2 (* l (/ (* (pow k -4) l) t))))

C

double code(double t, double l, double k) {
	return 2.0 * (l * ((pow(k, -4.0) * l) / t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (l * (((k ** (-4.0d0)) * l) / t))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (l * ((Math.pow(k, -4.0) * l) / t));
}

Python

def code(t, l, k):
	return 2.0 * (l * ((math.pow(k, -4.0) * l) / t))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(l * Float64(Float64((k ^ -4.0) * l) / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (l * (((k ^ -4.0) * l) / t));
end

Wolfram

code[t_, l_, k_] := N[(2 * N[(l * N[(N[(N[Power[k, -4], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 63.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 63.0%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. mult-flip N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   5. associate-*l* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

   12. pow-flip N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   13. lower-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   14. metadata-eval 69.0%

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right) \]

6. Applied rewrites 69.0%

   \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{-4}}{t}}\right)\right) \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{\color{blue}{t}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell \cdot {k}^{-4}}{\color{blue}{t}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell \cdot {k}^{-4}}{\color{blue}{t}}\right) \]

   5. *-commutative N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{{k}^{-4} \cdot \ell}{t}\right) \]

   6. lower-*.f64 69.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \frac{{k}^{-4} \cdot \ell}{t}\right) \]

8. Applied rewrites 69.7%

   \[\leadsto 2 \cdot \left(\ell \cdot \frac{{k}^{-4} \cdot \ell}{\color{blue}{t}}\right) \]
9. Add Preprocessing

Alternative 13: 69.1% accurate, 3.7× speedup

Math

\[\frac{\left(\ell + \ell\right) \cdot \left({k}^{-4} \cdot \ell\right)}{t} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (/ (* (+ l l) (* (pow k -4) l)) t))

C

double code(double t, double l, double k) {
	return ((l + l) * (pow(k, -4.0) * 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 = ((l + l) * ((k ** (-4.0d0)) * l)) / t
end function

Java

public static double code(double t, double l, double k) {
	return ((l + l) * (Math.pow(k, -4.0) * l)) / t;
}

Python

def code(t, l, k):
	return ((l + l) * (math.pow(k, -4.0) * l)) / t

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l + l) * Float64((k ^ -4.0) * l)) / t)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l + l) * ((k ^ -4.0) * l)) / t;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l + l), $MachinePrecision] * N[(N[Power[k, -4], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 63.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 63.0%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. mult-flip N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   5. associate-*l* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

   12. pow-flip N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   13. lower-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   14. metadata-eval 69.0%

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right) \]

6. Applied rewrites 69.0%

   \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{-4}}{t}}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{{k}^{-4}}{\color{blue}{t}}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell \cdot {k}^{-4}}{\color{blue}{t}} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot {k}^{-4}\right)}{\color{blue}{t}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot {k}^{-4}\right)}{\color{blue}{t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot {k}^{-4}\right)}{t} \]

   10. count-2-rev N/A

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \left(\ell \cdot {k}^{-4}\right)}{t} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \left(\ell \cdot {k}^{-4}\right)}{t} \]

   12. *-commutative N/A

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \left({k}^{-4} \cdot \ell\right)}{t} \]

   13. lower-*.f64 69.1%

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \left({k}^{-4} \cdot \ell\right)}{t} \]

8. Applied rewrites 69.1%

   \[\leadsto \frac{\left(\ell + \ell\right) \cdot \left({k}^{-4} \cdot \ell\right)}{\color{blue}{t}} \]
9. Add Preprocessing

Alternative 14: 69.0% accurate, 3.7× speedup

Math

\[\left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (+ l l) (* (/ (pow k -4) t) l)))

C

double code(double t, double l, double k) {
	return (l + l) * ((pow(k, -4.0) / t) * l);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l + l) * (((k ** (-4.0d0)) / t) * l)
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) * ((Math.pow(k, -4.0) / t) * l);
}

Python

def code(t, l, k):
	return (l + l) * ((math.pow(k, -4.0) / t) * l)

Julia

function code(t, l, k)
	return Float64(Float64(l + l) * Float64(Float64((k ^ -4.0) / t) * l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) * (((k ^ -4.0) / t) * l);
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] * N[(N[(N[Power[k, -4], $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 63.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 63.0%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. mult-flip N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   5. associate-*l* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

   12. pow-flip N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   13. lower-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   14. metadata-eval 69.0%

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right) \]

6. Applied rewrites 69.0%

   \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{-4}}{t}}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \color{blue}{\frac{{k}^{-4}}{t}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \color{blue}{\frac{{k}^{-4}}{t}} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{{k}^{-4}}}{t} \]

   8. count-2-rev N/A

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \frac{{\color{blue}{k}}^{-4}}{t} \]

   9. lower-+.f64 62.9%

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \frac{{\color{blue}{k}}^{-4}}{t} \]

8. Applied rewrites 62.9%

   \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \color{blue}{\frac{{k}^{-4}}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \color{blue}{\frac{{k}^{-4}}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{{k}^{-4}}}{t} \]

   3. associate-*l* N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

   5. *-commutative N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

   6. lower-*.f64 69.0%

      \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

10. Applied rewrites 69.0%

    \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\left(\frac{{k}^{-4}}{t} \cdot \ell\right)} \]
11. Add Preprocessing

Alternative 15: 64.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 29999999999999999061648523025317792442531696476907296615914263960963973225365257898969304001787672895785536451103497382835750166803163151679700391165952:\\ \;\;\;\;\left(\left(\left|\ell\right| + \left|\ell\right|\right) \cdot \left|\ell\right|\right) \cdot \frac{{k}^{-4}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \frac{k}{\left|\ell\right| \cdot \left|\ell\right|}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{1}}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<=
     (fabs l)
     29999999999999999061648523025317792442531696476907296615914263960963973225365257898969304001787672895785536451103497382835750166803163151679700391165952)
  (* (* (+ (fabs l) (fabs l)) (fabs l)) (/ (pow k -4) t))
  (/ 2 (/ (* (* k (/ k (* (fabs l) (fabs l)))) (* (- 1/2 1/2) t)) 1))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 3e+151) {
		tmp = ((fabs(l) + fabs(l)) * fabs(l)) * (pow(k, -4.0) / t);
	} else {
		tmp = 2.0 / (((k * (k / (fabs(l) * fabs(l)))) * ((0.5 - 0.5) * t)) / 1.0);
	}
	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 (abs(l) <= 3d+151) then
        tmp = ((abs(l) + abs(l)) * abs(l)) * ((k ** (-4.0d0)) / t)
    else
        tmp = 2.0d0 / (((k * (k / (abs(l) * abs(l)))) * ((0.5d0 - 0.5d0) * t)) / 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 3e+151) {
		tmp = ((Math.abs(l) + Math.abs(l)) * Math.abs(l)) * (Math.pow(k, -4.0) / t);
	} else {
		tmp = 2.0 / (((k * (k / (Math.abs(l) * Math.abs(l)))) * ((0.5 - 0.5) * t)) / 1.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 3e+151:
		tmp = ((math.fabs(l) + math.fabs(l)) * math.fabs(l)) * (math.pow(k, -4.0) / t)
	else:
		tmp = 2.0 / (((k * (k / (math.fabs(l) * math.fabs(l)))) * ((0.5 - 0.5) * t)) / 1.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 3e+151)
		tmp = Float64(Float64(Float64(abs(l) + abs(l)) * abs(l)) * Float64((k ^ -4.0) / t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / Float64(abs(l) * abs(l)))) * Float64(Float64(0.5 - 0.5) * t)) / 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 3e+151)
		tmp = ((abs(l) + abs(l)) * abs(l)) * ((k ^ -4.0) / t);
	else
		tmp = 2.0 / (((k * (k / (abs(l) * abs(l)))) * ((0.5 - 0.5) * t)) / 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 29999999999999999061648523025317792442531696476907296615914263960963973225365257898969304001787672895785536451103497382835750166803163151679700391165952], N[(N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, -4], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(k * N[(k / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1/2 - 1/2), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9999999999999999e151

   1. Initial program 36.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      5. lower-pow.f64 63.0%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   4. Applied rewrites 63.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

      13. lower-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

      14. metadata-eval 69.0%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right) \]

   6. Applied rewrites 69.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{-4}}{t}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \color{blue}{\frac{{k}^{-4}}{t}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \color{blue}{\frac{{k}^{-4}}{t}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot \ell\right) \cdot \frac{\color{blue}{{k}^{-4}}}{t} \]

      8. count-2-rev N/A

         \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \frac{{\color{blue}{k}}^{-4}}{t} \]

      9. lower-+.f64 62.9%

         \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \frac{{\color{blue}{k}}^{-4}}{t} \]

   8. Applied rewrites 62.9%

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \color{blue}{\frac{{k}^{-4}}{t}} \]

   if 2.9999999999999999e151 < l

   1. Initial program 36.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 73.6%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 73.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k}}} \]

   6. Applied rewrites 70.4%

      \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}{\color{blue}{\cos k}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{\cos k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.4%

      \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{\cos k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{1}} \]
   11. Step-by-step derivation

   12. Applied rewrites 34.3%

       \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 34.3% accurate, 8.1× speedup

Math

\[\frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{1}} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (/ 2 (/ (* (* k (/ k (* l l))) (* (- 1/2 1/2) t)) 1)))

C

double code(double t, double l, double k) {
	return 2.0 / (((k * (k / (l * l))) * ((0.5 - 0.5) * t)) / 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 / (((k * (k / (l * l))) * ((0.5d0 - 0.5d0) * t)) / 1.0d0)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((k * (k / (l * l))) * ((0.5 - 0.5) * t)) / 1.0);
}

Python

def code(t, l, k):
	return 2.0 / (((k * (k / (l * l))) * ((0.5 - 0.5) * t)) / 1.0)

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(k * Float64(k / Float64(l * l))) * Float64(Float64(0.5 - 0.5) * t)) / 1.0))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k * (k / (l * l))) * ((0.5 - 0.5) * t)) / 1.0);
end

Wolfram

code[t_, l_, k_] := N[(2 / N[(N[(N[(k * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1/2 - 1/2), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

   9. lower-cos.f64 73.6%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

4. Applied rewrites 73.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   4. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k}}} \]

6. Applied rewrites 70.4%

   \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}{\color{blue}{\cos k}}} \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{\cos k}} \]
8. Step-by-step derivation

9. Applied rewrites 36.4%

   \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{\cos k}} \]

10. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{1}} \]
11. Step-by-step derivation

12. Applied rewrites 34.3%

    \[\leadsto \frac{2}{\frac{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right)}{1}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))