Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 92.7%

Time: 18.3s

Alternatives: 16

Speedup: 6.6×

• 39.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 75.0

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

5. Applied rewrites 75.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. pow2 N/A

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

   13. frac-times N/A

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

   14. lower-/.f64 N/A

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

7. Applied rewrites 86.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 N/A

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

   22. lift-cos.f64 90.6

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

9. Applied rewrites 90.6%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-cos.f64 N/A

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

    6. associate-/l* N/A

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

    7. lower-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-/.f64 N/A

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

    11. *-commutative N/A

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

    12. lift-cos.f64 N/A

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

    13. lift-*.f64 93.0

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

11. Applied rewrites 93.0%

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

Alternative 2: 68.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \leq 10^{-46}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\ell} \cdot t}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (*
       (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
       (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))
      1e-46)
   (/
    2.0
    (*
     (* (* k k) t)
     (* (/ (fma (* k k) -0.3333333333333333 1.0) (* l l)) (* k k))))
   (/ 2.0 (/ (* (/ (pow k 4.0) l) t) l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0)) <= 1e-46) {
		tmp = 2.0 / (((k * k) * t) * ((fma((k * k), -0.3333333333333333, 1.0) / (l * l)) * (k * k)));
	} else {
		tmp = 2.0 / (((pow(k, 4.0) / l) * t) / l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (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)) <= 1e-46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t) * Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / Float64(l * l)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 4.0) / l) * t) / l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], 1e-46], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 1.00000000000000002e-46

   1. Initial program 85.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r/ N/A

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

       4. div-add-rev N/A

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

       5. +-commutative N/A

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

       6. lower-/.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 89.5

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)} \]

   11. Applied rewrites 89.5%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 1.00000000000000002e-46 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 9.6%

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

   3. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 59.9

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

   5. Applied rewrites 59.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 63.7

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

   7. Applied rewrites 63.7%

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

Alternative 3: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (- (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      -5e-8)
   (* -0.11666666666666667 (* l (/ l t)))
   (/ 2.0 (* (/ (* (* k k) (* k k)) l) (/ t l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0))) <= -5e-8) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = 2.0 / ((((k * k) * (k * k)) / l) * (t / 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 ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))) <= (-5d-8)) then
        tmp = (-0.11666666666666667d0) * (l * (l / t))
    else
        tmp = 2.0d0 / ((((k * k) * (k * k)) / l) * (t / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((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))) <= -5e-8) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = 2.0 / ((((k * k) * (k * k)) / l) * (t / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (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))) <= -5e-8:
		tmp = -0.11666666666666667 * (l * (l / t))
	else:
		tmp = 2.0 / ((((k * k) * (k * k)) / l) * (t / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= -5e-8)
		tmp = Float64(-0.11666666666666667 * Float64(l * Float64(l / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * Float64(k * k)) / l) * Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0))) <= -5e-8)
		tmp = -0.11666666666666667 * (l * (l / t));
	else
		tmp = 2.0 / ((((k * k) * (k * k)) / l) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], -5e-8], N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -4.9999999999999998e-8

   1. Initial program 80.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 5.7

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

   8. Applied rewrites 5.7%

      \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

         \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

      5. lower-/.f64 5.7

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

   10. Applied rewrites 5.7%

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

   if -4.9999999999999998e-8 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 32.0%

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

   3. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

Alternative 4: 54.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (- (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      -5e-8)
   (* -0.11666666666666667 (* l (/ l t)))
   (/ 2.0 (* (* (* k k) t) (/ (* k k) (* l l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0))) <= -5e-8) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))) <= (-5d-8)) then
        tmp = (-0.11666666666666667d0) * (l * (l / t))
    else
        tmp = 2.0d0 / (((k * k) * t) * ((k * k) / (l * l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((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))) <= -5e-8) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (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))) <= -5e-8:
		tmp = -0.11666666666666667 * (l * (l / t))
	else:
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= -5e-8)
		tmp = Float64(-0.11666666666666667 * Float64(l * Float64(l / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t) * Float64(Float64(k * k) / Float64(l * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0))) <= -5e-8)
		tmp = -0.11666666666666667 * (l * (l / t));
	else
		tmp = 2.0 / (((k * k) * t) * ((k * k) / (l * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], -5e-8], N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -4.9999999999999998e-8

   1. Initial program 80.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 5.7

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

   8. Applied rewrites 5.7%

      \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

         \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

      5. lower-/.f64 5.7

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

   10. Applied rewrites 5.7%

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

   if -4.9999999999999998e-8 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 32.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 63.7

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 63.5

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

   11. Applied rewrites 63.5%

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

Alternative 5: 54.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (- (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      -5e-8)
   (* -0.11666666666666667 (* l (/ l t)))
   (* (/ 2.0 (* (* k k) t)) (/ (* l l) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0))) <= -5e-8) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = (2.0 / ((k * k) * t)) * ((l * l) / (k * k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))) <= (-5d-8)) then
        tmp = (-0.11666666666666667d0) * (l * (l / t))
    else
        tmp = (2.0d0 / ((k * k) * t)) * ((l * l) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((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))) <= -5e-8) {
		tmp = -0.11666666666666667 * (l * (l / t));
	} else {
		tmp = (2.0 / ((k * k) * t)) * ((l * l) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (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))) <= -5e-8:
		tmp = -0.11666666666666667 * (l * (l / t))
	else:
		tmp = (2.0 / ((k * k) * t)) * ((l * l) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= -5e-8)
		tmp = Float64(-0.11666666666666667 * Float64(l * Float64(l / t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l * l) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0))) <= -5e-8)
		tmp = -0.11666666666666667 * (l * (l / t));
	else
		tmp = (2.0 / ((k * k) * t)) * ((l * l) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], -5e-8], N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -4.9999999999999998e-8

   1. Initial program 80.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 5.7

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

   8. Applied rewrites 5.7%

      \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

         \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

      5. lower-/.f64 5.7

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

   10. Applied rewrites 5.7%

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

   if -4.9999999999999998e-8 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 32.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.3

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

   8. Applied rewrites 63.3%

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

Alternative 6: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+238}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\ell} \cdot t}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.6e-6)
   (/
    2.0
    (*
     (* (/ (fma (* k k) -0.3333333333333333 1.0) l) (* k k))
     (* (/ (* k k) l) (/ t (cos k)))))
   (if (<= k 4.1e+238)
     (*
      (/ (/ 2.0 k) (* k t))
      (/ (* (cos k) (* l l)) (- 0.5 (* 0.5 (cos (* 2.0 k))))))
     (/ 2.0 (/ (* (/ (pow k 4.0) l) t) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.6e-6) {
		tmp = 2.0 / (((fma((k * k), -0.3333333333333333, 1.0) / l) * (k * k)) * (((k * k) / l) * (t / cos(k))));
	} else if (k <= 4.1e+238) {
		tmp = ((2.0 / k) / (k * t)) * ((cos(k) * (l * l)) / (0.5 - (0.5 * cos((2.0 * k)))));
	} else {
		tmp = 2.0 / (((pow(k, 4.0) / l) * t) / l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.6e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / l) * Float64(k * k)) * Float64(Float64(Float64(k * k) / l) * Float64(t / cos(k)))));
	elseif (k <= 4.1e+238)
		tmp = Float64(Float64(Float64(2.0 / k) / Float64(k * t)) * Float64(Float64(cos(k) * Float64(l * l)) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 4.0) / l) * t) / l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.6e-6], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.1e+238], N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.59999999999999984e-6

   1. Initial program 40.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 77.6

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   12. Applied rewrites 81.4%

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}} \]

   if 3.59999999999999984e-6 < k < 4.0999999999999999e238

   1. Initial program 27.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 89.4

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

   9. Applied rewrites 89.4%

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

   if 4.0999999999999999e238 < k

   1. Initial program 63.6%

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

   3. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 65.4

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

   7. Applied rewrites 65.4%

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

Alternative 7: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+238}:\\ \;\;\;\;\frac{2}{\left(k \cdot t\right) \cdot k} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\ell} \cdot t}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.6e-6)
   (/
    2.0
    (*
     (* (/ (fma (* k k) -0.3333333333333333 1.0) l) (* k k))
     (* (/ (* k k) l) (/ t (cos k)))))
   (if (<= k 3.1e+238)
     (*
      (/ 2.0 (* (* k t) k))
      (/ (* (cos k) (* l l)) (- 0.5 (* 0.5 (cos (* 2.0 k))))))
     (/ 2.0 (/ (* (/ (pow k 4.0) l) t) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.6e-6) {
		tmp = 2.0 / (((fma((k * k), -0.3333333333333333, 1.0) / l) * (k * k)) * (((k * k) / l) * (t / cos(k))));
	} else if (k <= 3.1e+238) {
		tmp = (2.0 / ((k * t) * k)) * ((cos(k) * (l * l)) / (0.5 - (0.5 * cos((2.0 * k)))));
	} else {
		tmp = 2.0 / (((pow(k, 4.0) / l) * t) / l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.6e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / l) * Float64(k * k)) * Float64(Float64(Float64(k * k) / l) * Float64(t / cos(k)))));
	elseif (k <= 3.1e+238)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * t) * k)) * Float64(Float64(cos(k) * Float64(l * l)) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 4.0) / l) * t) / l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.6e-6], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.1e+238], N[(N[(2.0 / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.59999999999999984e-6

   1. Initial program 40.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 77.6

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   12. Applied rewrites 81.4%

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}} \]

   if 3.59999999999999984e-6 < k < 3.10000000000000012e238

   1. Initial program 27.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 89.6

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

   9. Applied rewrites 89.6%

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

   if 3.10000000000000012e238 < k

   1. Initial program 63.6%

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

   3. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 65.4

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

   7. Applied rewrites 65.4%

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

Alternative 8: 80.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.6e-6)
   (/
    2.0
    (*
     (* (/ (fma (* k k) -0.3333333333333333 1.0) l) (* k k))
     (* (/ (* k k) l) (/ t (cos k)))))
   (/
    2.0
    (*
     (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) l)
     (/ (* (* k t) k) (* (cos k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.6e-6) {
		tmp = 2.0 / (((fma((k * k), -0.3333333333333333, 1.0) / l) * (k * k)) * (((k * k) / l) * (t / cos(k))));
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((2.0 * k)))) / l) * (((k * t) * k) / (cos(k) * l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.6e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / l) * Float64(k * k)) * Float64(Float64(Float64(k * k) / l) * Float64(t / cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / l) * Float64(Float64(Float64(k * t) * k) / Float64(cos(k) * l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.6e-6], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.59999999999999984e-6

   1. Initial program 40.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 77.6

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   12. Applied rewrites 81.4%

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}} \]

   if 3.59999999999999984e-6 < k

   1. Initial program 35.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. lift-cos.f64 89.9

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

   9. Applied rewrites 89.9%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 89.8

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

   11. Applied rewrites 89.8%

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

Alternative 9: 77.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.6e-6)
   (/
    2.0
    (*
     (* (/ (fma (* k k) -0.3333333333333333 1.0) l) (* k k))
     (* (/ (* k k) l) (/ t (cos k)))))
   (if (<= k 2.35e+138)
     (*
      (/ 2.0 (* (* k k) t))
      (/ (* (cos k) (* l l)) (- 0.5 (* 0.5 (cos (+ k k))))))
     (/ 2.0 (* (/ (pow (sin k) 2.0) l) (/ (* (* k t) k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.6e-6) {
		tmp = 2.0 / (((fma((k * k), -0.3333333333333333, 1.0) / l) * (k * k)) * (((k * k) / l) * (t / cos(k))));
	} else if (k <= 2.35e+138) {
		tmp = (2.0 / ((k * k) * t)) * ((cos(k) * (l * l)) / (0.5 - (0.5 * cos((k + k)))));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / l) * (((k * t) * k) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.6e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / l) * Float64(k * k)) * Float64(Float64(Float64(k * k) / l) * Float64(t / cos(k)))));
	elseif (k <= 2.35e+138)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(cos(k) * Float64(l * l)) / Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / l) * Float64(Float64(Float64(k * t) * k) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.6e-6], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.35e+138], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.59999999999999984e-6

   1. Initial program 40.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 77.6

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   12. Applied rewrites 81.4%

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}} \]

   if 3.59999999999999984e-6 < k < 2.3499999999999999e138

   1. Initial program 20.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 91.9

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

   7. Applied rewrites 91.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 91.9

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

   9. Applied rewrites 91.9%

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

   if 2.3499999999999999e138 < k

   1. Initial program 52.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. lift-cos.f64 83.4

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

   9. Applied rewrites 83.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative 67.2

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

   12. Applied rewrites 67.2%

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

Alternative 10: 73.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.3e+186)
   (/
    2.0
    (*
     (* (/ (fma (* k k) -0.3333333333333333 1.0) l) (* k k))
     (* (/ (* k k) l) (/ t (cos k)))))
   (* (/ 2.0 (* k (* k t))) (/ (* (cos k) (* l l)) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.3e+186) {
		tmp = 2.0 / (((fma((k * k), -0.3333333333333333, 1.0) / l) * (k * k)) * (((k * k) / l) * (t / cos(k))));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((cos(k) * (l * l)) / (k * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.3e+186)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / l) * Float64(k * k)) * Float64(Float64(Float64(k * k) / l) * Float64(t / cos(k)))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(cos(k) * Float64(l * l)) / Float64(k * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.3e+186], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e186

   1. Initial program 36.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 73.9

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   10. Applied rewrites 73.9%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, -0.3333333333333333, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   12. Applied rewrites 77.3%

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\cos k}\right)}} \]

   if 1.3e186 < l

   1. Initial program 65.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 88.5

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

   5. Applied rewrites 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. sqr-sin-a-rev N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

Alternative 11: 69.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\ell} \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.8e-171)
   (/ 2.0 (/ (* (/ (pow k 4.0) l) t) l))
   (* (/ 2.0 (* k (* k t))) (/ (* (cos k) (* l l)) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.8e-171) {
		tmp = 2.0 / (((pow(k, 4.0) / l) * t) / l);
	} else {
		tmp = (2.0 / (k * (k * t))) * ((cos(k) * (l * l)) / (k * k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 2.8d-171) then
        tmp = 2.0d0 / ((((k ** 4.0d0) / l) * t) / l)
    else
        tmp = (2.0d0 / (k * (k * t))) * ((cos(k) * (l * l)) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.8e-171) {
		tmp = 2.0 / (((Math.pow(k, 4.0) / l) * t) / l);
	} else {
		tmp = (2.0 / (k * (k * t))) * ((Math.cos(k) * (l * l)) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 2.8e-171:
		tmp = 2.0 / (((math.pow(k, 4.0) / l) * t) / l)
	else:
		tmp = (2.0 / (k * (k * t))) * ((math.cos(k) * (l * l)) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.8e-171)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 4.0) / l) * t) / l));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(cos(k) * Float64(l * l)) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 2.8e-171)
		tmp = 2.0 / ((((k ^ 4.0) / l) * t) / l);
	else
		tmp = (2.0 / (k * (k * t))) * ((cos(k) * (l * l)) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.8e-171], N[(2.0 / N[(N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.80000000000000023e-171

   1. Initial program 31.7%

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

   3. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

   if 2.80000000000000023e-171 < l

   1. Initial program 49.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 86.9

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lower-*.f64 90.4

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}{{\sin k}^{2}} \]

   7. Applied rewrites 90.4%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{2}}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}} \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k} \]

      4. lift-*.f64 75.7

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k} \]

   10. Applied rewrites 75.7%

       \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot \ell}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (/ (* k k) l) (/ (* (* k t) k) (* (cos k) l)))))

C

double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * (((k * t) * k) / (cos(k) * l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((k * k) / l) * (((k * t) * k) / (cos(k) * l)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * (((k * t) * k) / (Math.cos(k) * l)));
}

Python

def code(t, l, k):
	return 2.0 / (((k * k) / l) * (((k * t) * k) / (math.cos(k) * l)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(Float64(k * t) * k) / Float64(cos(k) * l))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k * k) / l) * (((k * t) * k) / (cos(k) * l)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 75.0

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 75.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\ell \cdot \ell}} \]

   5. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

   8. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k}}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\cos k}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot t}{\cos k}} \]

   13. frac-times N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left({k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left({k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

7. Applied rewrites 86.3%

   \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell \cdot \cos k}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   3. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \cos k}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \color{blue}{\cos k}}} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\ell \cdot \cos k}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \cos k}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \cos k}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(\color{blue}{k} \cdot k\right) \cdot t}{\ell \cdot \cos k}} \]

   11. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell \cdot \cos k}} \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell \cdot \cos k}} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \cos k}}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \cos k}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell} \cdot \cos k}} \]

   16. associate-*r* N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell} \cdot \cos k}} \]

   17. *-commutative N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\color{blue}{\ell} \cdot \cos k}} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\color{blue}{\ell} \cdot \cos k}} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \cos k}} \]

   20. *-commutative N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot \color{blue}{\ell}}} \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot \color{blue}{\ell}}} \]

   22. lift-cos.f64 90.6

       \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot \ell}} \]

9. Applied rewrites 90.6%

   \[\leadsto \frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \color{blue}{\frac{\left(k \cdot t\right) \cdot k}{\cos k \cdot \ell}}} \]

10. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\frac{{k}^{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot t\right)} \cdot k}{\cos k \cdot \ell}} \]
11. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{\left(k \cdot \color{blue}{t}\right) \cdot k}{\cos k \cdot \ell}} \]

    2. lift-*.f64 78.1

       \[\leadsto \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{\left(k \cdot \color{blue}{t}\right) \cdot k}{\cos k \cdot \ell}} \]

12. Applied rewrites 78.1%

    \[\leadsto \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\left(k \cdot t\right)} \cdot k}{\cos k \cdot \ell}} \]
13. Add Preprocessing

Alternative 13: 68.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-298}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 4e-298)
   (/ 2.0 (* (/ (pow k 4.0) l) (/ t l)))
   (/
    2.0
    (*
     (* (* k k) t)
     (* (/ (fma (* k k) -0.3333333333333333 1.0) (* l l)) (* k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 4e-298) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t / l));
	} else {
		tmp = 2.0 / (((k * k) * t) * ((fma((k * k), -0.3333333333333333, 1.0) / (l * l)) * (k * k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 4e-298)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t) * Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / Float64(l * l)) * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e-298], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 3.99999999999999965e-298

   1. Initial program 20.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \color{blue}{\ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{\color{blue}{t}}{\ell}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}} \]

      6. lower-/.f64 69.5

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\color{blue}{\ell}}} \]

   5. Applied rewrites 69.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]

   if 3.99999999999999965e-298 < (*.f64 l l)

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 83.2

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 83.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      3. lift-*.f64 71.8

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   8. Applied rewrites 71.8%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right)}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(\frac{\frac{-1}{3} \cdot {k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) \cdot {k}^{2}\right)} \]

       4. div-add-rev N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\frac{-1}{3} \cdot {k}^{2} + 1}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1 + \frac{-1}{3} \cdot {k}^{2}}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1 + \frac{-1}{3} \cdot {k}^{2}}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\frac{-1}{3} \cdot {k}^{2} + 1}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{{k}^{2} \cdot \frac{-1}{3} + 1}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left({k}^{2}, \frac{-1}{3}, 1\right)}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       12. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{\ell \cdot \ell} \cdot {k}^{2}\right)} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{\ell \cdot \ell} \cdot {k}^{2}\right)} \]

       14. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)} \]

       15. lift-*.f64 72.4

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)} \]

   11. Applied rewrites 72.4%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 8e-155)
   (/ 2.0 (* (/ (* (* k k) (* k k)) l) (/ t l)))
   (/
    2.0
    (*
     (* (* k k) t)
     (* (/ (fma (* k k) -0.3333333333333333 1.0) (* l l)) (* k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 8e-155) {
		tmp = 2.0 / ((((k * k) * (k * k)) / l) * (t / l));
	} else {
		tmp = 2.0 / (((k * k) * t) * ((fma((k * k), -0.3333333333333333, 1.0) / (l * l)) * (k * k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 8e-155)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * Float64(k * k)) / l) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t) * Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 1.0) / Float64(l * l)) * Float64(k * k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 8e-155], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.00000000000000011e-155

   1. Initial program 31.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \color{blue}{\ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{\color{blue}{t}}{\ell}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}} \]

      6. lower-/.f64 67.9

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\color{blue}{\ell}}} \]

   5. Applied rewrites 67.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{{k}^{\left(2 + 2\right)}}{\ell} \cdot \frac{t}{\ell}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {k}^{2}}{\ell} \cdot \frac{t}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {k}^{2}}{\ell} \cdot \frac{t}{\ell}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot {k}^{2}}{\ell} \cdot \frac{t}{\ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot {k}^{2}}{\ell} \cdot \frac{t}{\ell}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t}{\ell}} \]

      8. lift-*.f64 67.9

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t}{\ell}} \]

   7. Applied rewrites 67.9%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t}{\ell}} \]

   if 8.00000000000000011e-155 < l

   1. Initial program 50.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 87.8

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 87.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      3. lift-*.f64 73.1

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   8. Applied rewrites 73.1%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right)}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

       3. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(\frac{\frac{-1}{3} \cdot {k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) \cdot {k}^{2}\right)} \]

       4. div-add-rev N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\frac{-1}{3} \cdot {k}^{2} + 1}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1 + \frac{-1}{3} \cdot {k}^{2}}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1 + \frac{-1}{3} \cdot {k}^{2}}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\frac{-1}{3} \cdot {k}^{2} + 1}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{{k}^{2} \cdot \frac{-1}{3} + 1}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left({k}^{2}, \frac{-1}{3}, 1\right)}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{{\ell}^{2}} \cdot {k}^{2}\right)} \]

       12. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{\ell \cdot \ell} \cdot {k}^{2}\right)} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{\ell \cdot \ell} \cdot {k}^{2}\right)} \]

       14. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, \frac{-1}{3}, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)} \]

       15. lift-*.f64 74.3

           \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right)} \]

   11. Applied rewrites 74.3%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 1\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 20.2% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (/ (* l l) t)))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * ((l * l) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * ((l * l) / t)
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * ((l * l) / t);
}

Python

def code(t, l, k):
	return -0.11666666666666667 * ((l * l) / t)

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(Float64(l * l) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * ((l * l) / t);
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 31.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-/.f64 18.2

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 18.2%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Add Preprocessing

Alternative 16: 18.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

def code(t, l, k):
	return -0.11666666666666667 * (l * (l / t))

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(l * Float64(l / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * (l * (l / t));
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 31.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-/.f64 18.2

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 18.2%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 16.0

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

10. Applied rewrites 16.0%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025043 
(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))))