Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 98.6%

Time: 8.2s

Alternatives: 8

Speedup: 5.8×

• 39.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in t around 0

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

4. Applied rewrites 72.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. pow2 N/A

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

   8. associate-*r/ N/A

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

   9. pow2 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-tan.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-sin.f64 83.2

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

6. Applied rewrites 83.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

8. Applied rewrites 86.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

10. Applied rewrites 98.6%

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

Alternative 2: 80.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k \cdot k}{\ell}\\ \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(t\_1 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(\sin k \cdot t\right) \cdot \left(\frac{k}{\ell \cdot \ell} \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* k k) l)))
   (if (<= l 1.8e-163)
     (/ 2.0 (* t_1 (* t_1 t)))
     (/ 2.0 (* (tan k) (* (* (sin k) t) (* (/ k (* l l)) k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k * k) / l;
	double tmp;
	if (l <= 1.8e-163) {
		tmp = 2.0 / (t_1 * (t_1 * t));
	} else {
		tmp = 2.0 / (tan(k) * ((sin(k) * t) * ((k / (l * l)) * k)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k * k) / l
    if (l <= 1.8d-163) then
        tmp = 2.0d0 / (t_1 * (t_1 * t))
    else
        tmp = 2.0d0 / (tan(k) * ((sin(k) * t) * ((k / (l * l)) * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k * k) / l;
	double tmp;
	if (l <= 1.8e-163) {
		tmp = 2.0 / (t_1 * (t_1 * t));
	} else {
		tmp = 2.0 / (Math.tan(k) * ((Math.sin(k) * t) * ((k / (l * l)) * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k * k) / l
	tmp = 0
	if l <= 1.8e-163:
		tmp = 2.0 / (t_1 * (t_1 * t))
	else:
		tmp = 2.0 / (math.tan(k) * ((math.sin(k) * t) * ((k / (l * l)) * k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k * k) / l)
	tmp = 0.0
	if (l <= 1.8e-163)
		tmp = Float64(2.0 / Float64(t_1 * Float64(t_1 * t)));
	else
		tmp = Float64(2.0 / Float64(tan(k) * Float64(Float64(sin(k) * t) * Float64(Float64(k / Float64(l * l)) * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k * k) / l;
	tmp = 0.0;
	if (l <= 1.8e-163)
		tmp = 2.0 / (t_1 * (t_1 * t));
	else
		tmp = 2.0 / (tan(k) * ((sin(k) * t) * ((k / (l * l)) * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[l, 1.8e-163], N[(2.0 / N[(t$95$1 * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * N[(N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.7999999999999999e-163

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 62.9

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

   4. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 74.4

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

   6. Applied rewrites 74.4%

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

   if 1.7999999999999999e-163 < l

   1. Initial program 36.2%

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

   3. Applied rewrites 30.9%

      \[\leadsto \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 0\right)\right)}} \]

   4. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 77.4

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

   6. Applied rewrites 77.4%

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

Alternative 3: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell + \ell}{k \cdot k} \cdot \frac{\frac{\ell}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-48)
   (* (/ (+ l l) (* k k)) (/ (/ l (* k k)) t))
   (/ 2.0 (* k (* k (* (/ t (* l l)) (* (tan k) (sin k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-48) {
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t);
	} else {
		tmp = 2.0 / (k * (k * ((t / (l * l)) * (tan(k) * sin(k)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.15d-48) then
        tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t)
    else
        tmp = 2.0d0 / (k * (k * ((t / (l * l)) * (tan(k) * sin(k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-48) {
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t);
	} else {
		tmp = 2.0 / (k * (k * ((t / (l * l)) * (Math.tan(k) * Math.sin(k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.15e-48:
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t)
	else:
		tmp = 2.0 / (k * (k * ((t / (l * l)) * (math.tan(k) * math.sin(k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.15e-48)
		tmp = Float64(Float64(Float64(l + l) / Float64(k * k)) * Float64(Float64(l / Float64(k * k)) / t));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t / Float64(l * l)) * Float64(tan(k) * sin(k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.15e-48)
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t);
	else
		tmp = 2.0 / (k * (k * ((t / (l * l)) * (tan(k) * sin(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-48], N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-48

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. pow-plus N/A

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

      24. associate-*l* N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 63.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/l/ N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 74.4%

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{t}} \]

   if 1.15e-48 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 75.0%

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

Alternative 4: 74.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+186}:\\ \;\;\;\;\frac{\ell + \ell}{k \cdot k} \cdot \frac{\frac{\ell}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot \left(k \cdot \left(\frac{k}{\ell \cdot \ell} \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.8e+186)
   (* (/ (+ l l) (* k k)) (/ (/ l (* k k)) t))
   (/ 2.0 (* t (* k (* k (* (/ k (* l l)) k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.8e+186) {
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t);
	} else {
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.8d+186) then
        tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t)
    else
        tmp = 2.0d0 / (t * (k * (k * ((k / (l * l)) * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.8e+186) {
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t);
	} else {
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.8e+186:
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t)
	else:
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.8e+186)
		tmp = Float64(Float64(Float64(l + l) / Float64(k * k)) * Float64(Float64(l / Float64(k * k)) / t));
	else
		tmp = Float64(2.0 / Float64(t * Float64(k * Float64(k * Float64(Float64(k / Float64(l * l)) * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.8e+186)
		tmp = ((l + l) / (k * k)) * ((l / (k * k)) / t);
	else
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.8e+186], N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(k * N[(k * N[(N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8000000000000001e186

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. pow-plus N/A

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

      24. associate-*l* N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 63.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. associate-/l/ N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 74.4%

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{t}} \]

   if 1.8000000000000001e186 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 62.9

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

   4. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-*l/ N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 65.0

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

   6. Applied rewrites 65.0%

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

Alternative 5: 72.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot \left(k \cdot \left(\frac{k}{\ell \cdot \ell} \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-48)
   (* (/ (+ l l) (* k k)) (/ l (* (* k k) t)))
   (/ 2.0 (* t (* k (* k (* (/ k (* l l)) k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-48) {
		tmp = ((l + l) / (k * k)) * (l / ((k * k) * t));
	} else {
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.15d-48) then
        tmp = ((l + l) / (k * k)) * (l / ((k * k) * t))
    else
        tmp = 2.0d0 / (t * (k * (k * ((k / (l * l)) * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.15e-48) {
		tmp = ((l + l) / (k * k)) * (l / ((k * k) * t));
	} else {
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.15e-48:
		tmp = ((l + l) / (k * k)) * (l / ((k * k) * t))
	else:
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.15e-48)
		tmp = Float64(Float64(Float64(l + l) / Float64(k * k)) * Float64(l / Float64(Float64(k * k) * t)));
	else
		tmp = Float64(2.0 / Float64(t * Float64(k * Float64(k * Float64(Float64(k / Float64(l * l)) * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.15e-48)
		tmp = ((l + l) / (k * k)) * (l / ((k * k) * t));
	else
		tmp = 2.0 / (t * (k * (k * ((k / (l * l)) * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-48], N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(k * N[(k * N[(N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-48

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. metadata-eval N/A

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

      23. pow-plus N/A

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

      24. associate-*l* N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 63.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. pow2 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.9

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

   8. Applied rewrites 72.9%

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

   if 1.15e-48 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 62.9

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

   4. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-*l/ N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 65.0

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

   6. Applied rewrites 65.0%

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

Alternative 6: 72.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.8

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

4. Applied rewrites 62.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow-prod-down N/A

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

   10. pow-sqr N/A

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

   11. metadata-eval N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

   14. pow2 N/A

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

   15. *-commutative N/A

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

   16. count-2-rev N/A

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

   17. pow2 N/A

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

   18. pow2 N/A

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

   19. distribute-rgt-out N/A

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

   20. lower-*.f64 N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval N/A

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

   23. pow-plus N/A

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

   24. associate-*l* N/A

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

   25. lower-*.f64 N/A

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

6. Applied rewrites 63.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. associate-*l* N/A

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

   12. associate-*l* N/A

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

   13. pow2 N/A

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

   14. associate-*l* N/A

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

   15. pow-prod-up N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

8. Applied rewrites 68.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. pow2 N/A

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

   22. lift-*.f64 71.7

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

10. Applied rewrites 71.7%

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

Alternative 7: 71.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.8

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

4. Applied rewrites 62.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow-prod-down N/A

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

   10. pow-sqr N/A

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

   11. metadata-eval N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

   14. pow2 N/A

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

   15. *-commutative N/A

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

   16. count-2-rev N/A

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

   17. pow2 N/A

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

   18. pow2 N/A

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

   19. distribute-rgt-out N/A

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

   20. lower-*.f64 N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval N/A

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

   23. pow-plus N/A

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

   24. associate-*l* N/A

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

   25. lower-*.f64 N/A

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

6. Applied rewrites 63.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. pow2 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.9

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

8. Applied rewrites 72.9%

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

Alternative 8: 70.3% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.8

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

4. Applied rewrites 62.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow-prod-down N/A

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

   10. pow-sqr N/A

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

   11. metadata-eval N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

   14. pow2 N/A

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

   15. *-commutative N/A

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

   16. count-2-rev N/A

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

   17. pow2 N/A

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

   18. pow2 N/A

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

   19. distribute-rgt-out N/A

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

   20. lower-*.f64 N/A

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

   21. lower-+.f64 N/A

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

   22. metadata-eval N/A

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

   23. pow-plus N/A

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

   24. associate-*l* N/A

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

   25. lower-*.f64 N/A

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

6. Applied rewrites 63.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. associate-*l* N/A

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

   12. associate-*l* N/A

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

   13. pow2 N/A

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

   14. associate-*l* N/A

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

   15. pow-prod-up N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

8. Applied rewrites 68.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 70.3

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

10. Applied rewrites 70.3%

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

Reproduce

herbie shell --seed 2025129 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))