Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 98.6%

Time: 7.3s

Alternatives: 6

Speedup: 6.0×

• 59.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (t l k)
  :precision binary64
  (/
 2.0
 (*
  (* (* (/ (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: 36.0% accurate, 1.0× speedup

Math

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

FPCore

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-cos.f64 74.2%

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

4. Applied rewrites 74.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*r/ N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. tan-quot N/A

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

   15. lift-tan.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 77.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 91.2%

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

8. Applied rewrites 91.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 98.6%

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

10. Applied rewrites 98.6%

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

Alternative 2: 85.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 1.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\ell + \ell}{\left|k\right| \cdot \left|k\right|} \cdot \frac{\ell}{\left(\left|k\right| \cdot t\right) \cdot \left|k\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin \left(\left|k\right|\right) \cdot \left(t \cdot \left(\frac{\left|k\right|}{\ell \cdot \ell} \cdot \tan \left(\left|k\right|\right)\right)\right)\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 1.2e-52)
  (*
   (/ (+ l l) (* (fabs k) (fabs k)))
   (/ l (* (* (fabs k) t) (fabs k))))
  (/
   2.0
   (*
    (* (sin (fabs k)) (* t (* (/ (fabs k) (* l l)) (tan (fabs k)))))
    (fabs k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.2e-52) {
		tmp = ((l + l) / (fabs(k) * fabs(k))) * (l / ((fabs(k) * t) * fabs(k)));
	} else {
		tmp = 2.0 / ((sin(fabs(k)) * (t * ((fabs(k) / (l * l)) * tan(fabs(k))))) * fabs(k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 1.2d-52) then
        tmp = ((l + l) / (abs(k) * abs(k))) * (l / ((abs(k) * t) * abs(k)))
    else
        tmp = 2.0d0 / ((sin(abs(k)) * (t * ((abs(k) / (l * l)) * tan(abs(k))))) * abs(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 1.2e-52) {
		tmp = ((l + l) / (Math.abs(k) * Math.abs(k))) * (l / ((Math.abs(k) * t) * Math.abs(k)));
	} else {
		tmp = 2.0 / ((Math.sin(Math.abs(k)) * (t * ((Math.abs(k) / (l * l)) * Math.tan(Math.abs(k))))) * Math.abs(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1.2e-52:
		tmp = ((l + l) / (math.fabs(k) * math.fabs(k))) * (l / ((math.fabs(k) * t) * math.fabs(k)))
	else:
		tmp = 2.0 / ((math.sin(math.fabs(k)) * (t * ((math.fabs(k) / (l * l)) * math.tan(math.fabs(k))))) * math.fabs(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.2e-52)
		tmp = Float64(Float64(Float64(l + l) / Float64(abs(k) * abs(k))) * Float64(l / Float64(Float64(abs(k) * t) * abs(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(abs(k)) * Float64(t * Float64(Float64(abs(k) / Float64(l * l)) * tan(abs(k))))) * abs(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1.2e-52)
		tmp = ((l + l) / (abs(k) * abs(k))) * (l / ((abs(k) * t) * abs(k)));
	else
		tmp = 2.0 / ((sin(abs(k)) * (t * ((abs(k) / (l * l)) * tan(abs(k))))) * abs(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1.2e-52], N[(N[(N[(l + l), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(t * N[(N[(N[Abs[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.2000000000000001e-52

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.6%

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 68.6%

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

   6. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{4}}}{t} \]

      11. sqr-pow N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      13. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      14. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      15. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      16. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      17. associate-/r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{t}} \]

   8. Applied rewrites 63.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.2%

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

   10. Applied rewrites 73.2%

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

   if 1.2000000000000001e-52 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 74.2%

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

   4. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. tan-quot N/A

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

      15. lift-tan.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.3%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 79.6%

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

   8. Applied rewrites 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.6%

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

   10. Applied rewrites 79.6%

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

Alternative 3: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 4.8 \cdot 10^{+179}:\\ \;\;\;\;\frac{\left|\ell\right| + \left|\ell\right|}{k \cdot k} \cdot \frac{\left|\ell\right|}{\left(k \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5\right)}{\cos k}}{\left|\ell\right| \cdot \left|\ell\right|}}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs l) 4.8e+179)
  (* (/ (+ (fabs l) (fabs l)) (* k k)) (/ (fabs l) (* (* k t) k)))
  (/
   2.0
   (/
    (/ (* (* (* k k) t) (- 0.5 0.5)) (cos k))
    (* (fabs l) (fabs l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 4.8e+179) {
		tmp = ((fabs(l) + fabs(l)) / (k * k)) * (fabs(l) / ((k * t) * k));
	} else {
		tmp = 2.0 / (((((k * k) * t) * (0.5 - 0.5)) / cos(k)) / (fabs(l) * fabs(l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(l) <= 4.8d+179) then
        tmp = ((abs(l) + abs(l)) / (k * k)) * (abs(l) / ((k * t) * k))
    else
        tmp = 2.0d0 / (((((k * k) * t) * (0.5d0 - 0.5d0)) / cos(k)) / (abs(l) * abs(l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 4.8e+179) {
		tmp = ((Math.abs(l) + Math.abs(l)) / (k * k)) * (Math.abs(l) / ((k * t) * k));
	} else {
		tmp = 2.0 / (((((k * k) * t) * (0.5 - 0.5)) / Math.cos(k)) / (Math.abs(l) * Math.abs(l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 4.8e+179:
		tmp = ((math.fabs(l) + math.fabs(l)) / (k * k)) * (math.fabs(l) / ((k * t) * k))
	else:
		tmp = 2.0 / (((((k * k) * t) * (0.5 - 0.5)) / math.cos(k)) / (math.fabs(l) * math.fabs(l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 4.8e+179)
		tmp = Float64(Float64(Float64(abs(l) + abs(l)) / Float64(k * k)) * Float64(abs(l) / Float64(Float64(k * t) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * Float64(0.5 - 0.5)) / cos(k)) / Float64(abs(l) * abs(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 4.8e+179)
		tmp = ((abs(l) + abs(l)) / (k * k)) * (abs(l) / ((k * t) * k));
	else
		tmp = 2.0 / (((((k * k) * t) * (0.5 - 0.5)) / cos(k)) / (abs(l) * abs(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 4.8e+179], N[(N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.8000000000000003e179

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.6%

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 68.6%

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

   6. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{4}}}{t} \]

      11. sqr-pow N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      13. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      14. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      15. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      16. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      17. associate-/r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{t}} \]

   8. Applied rewrites 63.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.2%

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

   10. Applied rewrites 73.2%

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

   if 4.8000000000000003e179 < l

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 74.2%

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

   4. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - \frac{1}{2}\right)}{\cos k}}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.1%

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

Alternative 4: 73.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{-76}:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \frac{\ell}{t\_1}}{\left(k \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{t\_1 \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* k (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1e-76)
     (/ (* (+ l l) (/ l t_1)) (* (* k k) k))
     (* (/ (+ l l) (* k k)) (/ l (* t_1 k)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 1e-76) {
		tmp = ((l + l) * (l / t_1)) / ((k * k) * k);
	} else {
		tmp = ((l + l) / (k * k)) * (l / (t_1 * k));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1e-76) {
		tmp = ((l + l) * (l / t_1)) / ((k * k) * k);
	} else {
		tmp = ((l + l) / (k * k)) * (l / (t_1 * k));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1e-76:
		tmp = ((l + l) * (l / t_1)) / ((k * k) * k)
	else:
		tmp = ((l + l) / (k * k)) * (l / (t_1 * k))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 1e-76)
		tmp = Float64(Float64(Float64(l + l) * Float64(l / t_1)) / Float64(Float64(k * k) * k));
	else
		tmp = Float64(Float64(Float64(l + l) / Float64(k * k)) * Float64(l / Float64(t_1 * k)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1e-76)
		tmp = ((l + l) * (l / t_1)) / ((k * k) * k);
	else
		tmp = ((l + l) / (k * k)) * (l / (t_1 * k));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e-76], N[(N[(N[(l + l), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$1 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999993e-77

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.6%

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 68.6%

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

   6. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{4}}}{t} \]

      11. sqr-pow N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      13. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      14. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      15. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      16. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      17. associate-/r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{t}} \]

   8. Applied rewrites 63.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 70.0%

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

   10. Applied rewrites 70.0%

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

   if 9.9999999999999993e-77 < t

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.6%

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

   4. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 68.6%

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

   6. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{4}}}{t} \]

      11. sqr-pow N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      13. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      14. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      15. lower-unsound-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      16. lower-unsound-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

      17. associate-/r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{t}} \]

   8. Applied rewrites 63.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.2%

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

   10. Applied rewrites 73.2%

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

Alternative 5: 73.2% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 62.6%

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

4. Applied rewrites 62.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   5. associate-*l* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval 68.6%

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

6. Applied rewrites 68.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. pow-flip N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{4}}}{t} \]

   11. sqr-pow N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   12. lower-unsound-*.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   13. lower-unsound-pow.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   14. lower-unsound-/.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   15. lower-unsound-pow.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   16. lower-unsound-/.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   17. associate-/r* N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}} \]

   18. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{t}} \]

8. Applied rewrites 63.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 73.2%

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

10. Applied rewrites 73.2%

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

Alternative 6: 70.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

2. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 62.6%

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

4. Applied rewrites 62.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   5. associate-*l* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval 68.6%

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

6. Applied rewrites 68.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. pow-flip N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{4}}}{t} \]

   11. sqr-pow N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   12. lower-unsound-*.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   13. lower-unsound-pow.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   14. lower-unsound-/.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   15. lower-unsound-pow.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   16. lower-unsound-/.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{1}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{t} \]

   17. associate-/r* N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}} \]

   18. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{t}} \]

8. Applied rewrites 63.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 70.0%

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 70.0%

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

10. Applied rewrites 70.0%

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

Reproduce

herbie shell --seed 2025214 
(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))))