Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 97.7%

Time: 12.8s

Alternatives: 19

Speedup: 4.4×

• 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: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

2. 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. Applied rewrites 74.0%

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

6. Applied rewrites 74.5%

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

8. Applied rewrites 85.8%

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

10. Applied rewrites 97.7%

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

Alternative 2: 76.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{0.5 - 0.5 \cdot \cos \left(k + k\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    2.0
    (* (/ l k) (* (/ l k) (/ (/ (cos k) t) (- 0.5 (* 0.5 (cos (+ k k))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t) / (0.5 - (0.5 * cos((k + k)))))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(cos(k) / t) / Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   10. Applied rewrites 85.8%

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

Alternative 3: 76.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell + \ell}{k} \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}\right) \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    (* (/ (+ l l) k) (/ (cos k) (* (- 0.5 (* 0.5 (cos (+ k k)))) t)))
    (/ l k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = (((l + l) / k) * (cos(k) / ((0.5 - (0.5 * cos((k + k)))) * t))) * (l / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / k) * Float64(cos(k) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t))) * Float64(l / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   10. Applied rewrites 85.8%

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

Alternative 4: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\cos k \cdot \ell}{\left(k \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    (/ (+ l l) k)
    (/ (* (cos k) l) (* (* k t) (- 0.5 (* 0.5 (cos (+ k k)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = ((l + l) / k) * ((cos(k) * l) / ((k * t) * (0.5 - (0.5 * cos((k + k))))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(l + l) / k) * Float64(Float64(cos(k) * l) / Float64(Float64(k * t) * Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   10. Applied rewrites 82.8%

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

Alternative 5: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot \left(k \cdot t\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    2.0
    (* l (* l (/ (cos k) (* (* (- 0.5 (* 0.5 (cos (+ k k)))) k) (* k t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = 2.0 * (l * (l * (cos(k) / (((0.5 - (0.5 * cos((k + k)))) * k) * (k * t)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * k) * Float64(k * t))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 77.8%

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

Alternative 6: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right)\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    2.0
    (* l (* l (/ (cos k) (* k (* k (* (- 0.5 (* 0.5 (cos (+ k k)))) t)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = 2.0 * (l * (l * (cos(k) / (k * (k * ((0.5 - (0.5 * cos((k + k)))) * t))))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(k * Float64(k * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t)))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(k * N[(k * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 77.8%

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

Alternative 7: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \cos k}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (/
    (* (* (+ l l) l) (cos k))
    (* (* (* k t) k) (- 0.5 (* 0.5 (cos (+ k k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = (((l + l) * l) * cos(k)) / (((k * t) * k) * (0.5 - (0.5 * cos((k + k)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) * cos(k)) / Float64(Float64(Float64(k * t) * k) * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 67.2%

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

   10. Applied rewrites 69.6%

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

Alternative 8: 73.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\ell + \ell\right) \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    (* (+ l l) (/ (cos k) (* (* (* k k) t) (- 0.5 (* 0.5 (cos (+ k k)))))))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = ((l + l) * (cos(k) / (((k * k) * t) * (0.5 - (0.5 * cos((k + k))))))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(l + l) * Float64(cos(k) / Float64(Float64(Float64(k * k) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 74.5%

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

Alternative 9: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \left(\ell + \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    (/ (* (cos k) l) (* (* (* k k) t) (- 0.5 (* 0.5 (cos (+ k k))))))
    (+ l l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = ((cos(k) * l) / (((k * k) * t) * (0.5 - (0.5 * cos((k + k)))))) * (l + l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(cos(k) * l) / Float64(Float64(Float64(k * k) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))))) * Float64(l + l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 74.5%

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

Alternative 10: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.28 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \left(\left(\ell + \ell\right) \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.28e-8)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    (/ (cos k) (* (* (* k k) t) (- 0.5 (* 0.5 (cos (+ k k))))))
    (* (+ l l) l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.28e-8) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = (cos(k) / (((k * k) * t) * (0.5 - (0.5 * cos((k + k)))))) * ((l + l) * l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.28e-8)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64(k * k) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))))) * Float64(Float64(l + l) * l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.28e-8], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.28000000000000005e-8

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 1.28000000000000005e-8 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 67.2%

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

Alternative 11: 71.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.9 \cdot 10^{+69}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{k \cdot \left(\left(k \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.9e+69)
   (*
    2.0
    (*
     (/ l k)
     (*
      (/ l k)
      (/ (fma -0.16666666666666666 (/ (pow k 2.0) t) (/ 1.0 t)) (pow k 2.0)))))
   (*
    2.0
    (* l (* l (/ 1.0 (* k (* (* k t) (- 0.5 (* 0.5 (cos (+ k k))))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.9e+69) {
		tmp = 2.0 * ((l / k) * ((l / k) * (fma(-0.16666666666666666, (pow(k, 2.0) / t), (1.0 / t)) / pow(k, 2.0))));
	} else {
		tmp = 2.0 * (l * (l * (1.0 / (k * ((k * t) * (0.5 - (0.5 * cos((k + k)))))))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.9e+69)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(fma(-0.16666666666666666, Float64((k ^ 2.0) / t), Float64(1.0 / t)) / (k ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(1.0 / Float64(k * Float64(Float64(k * t) * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.9e+69], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(1.0 / N[(k * N[(N[(k * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.8999999999999999e69

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{2}}}\right)\right) \]

   11. Applied rewrites 57.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{\color{blue}{{k}^{2}}}\right)\right) \]

   if 3.8999999999999999e69 < 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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 64.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+124}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{\left(0.5 - 0.5\right) \cdot t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 7.5e+124)
   (* 2.0 (* (/ l k) (* (/ l k) (/ 1.0 (* (pow k 2.0) t)))))
   (* 2.0 (* (/ l k) (* (/ l k) (/ (cos k) (* (- 0.5 0.5) t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 7.5e+124) {
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (pow(k, 2.0) * t))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) / ((0.5 - 0.5) * t))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 7.5d+124) then
        tmp = 2.0d0 * ((l / k) * ((l / k) * (1.0d0 / ((k ** 2.0d0) * t))))
    else
        tmp = 2.0d0 * ((l / k) * ((l / k) * (cos(k) / ((0.5d0 - 0.5d0) * t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 7.5e+124) {
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (Math.pow(k, 2.0) * t))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * (Math.cos(k) / ((0.5 - 0.5) * t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 7.5e+124:
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (math.pow(k, 2.0) * t))))
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * (math.cos(k) / ((0.5 - 0.5) * t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 7.5e+124)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(1.0 / Float64((k ^ 2.0) * t)))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(cos(k) / Float64(Float64(0.5 - 0.5) * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 7.5e+124)
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / ((k ^ 2.0) * t))));
	else
		tmp = 2.0 * ((l / k) * ((l / k) * (cos(k) / ((0.5 - 0.5) * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 7.5e+124], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.50000000000000038e124

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 73.0%

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

   if 7.50000000000000038e124 < l

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 41.3%

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

Alternative 13: 71.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.7 \cdot 10^{+128}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot \ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 6.7e+128)
   (* 2.0 (* (/ l k) (* (/ l k) (/ 1.0 (* (pow k 2.0) t)))))
   (* (* 2.0 (/ (* l l) (* k k))) (/ (cos k) (* t (- 0.5 0.5))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.7e+128) {
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (pow(k, 2.0) * t))));
	} else {
		tmp = (2.0 * ((l * l) / (k * k))) * (cos(k) / (t * (0.5 - 0.5)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 6.7d+128) then
        tmp = 2.0d0 * ((l / k) * ((l / k) * (1.0d0 / ((k ** 2.0d0) * t))))
    else
        tmp = (2.0d0 * ((l * l) / (k * k))) * (cos(k) / (t * (0.5d0 - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.7e+128) {
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (Math.pow(k, 2.0) * t))));
	} else {
		tmp = (2.0 * ((l * l) / (k * k))) * (Math.cos(k) / (t * (0.5 - 0.5)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 6.7e+128:
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (math.pow(k, 2.0) * t))))
	else:
		tmp = (2.0 * ((l * l) / (k * k))) * (math.cos(k) / (t * (0.5 - 0.5)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 6.7e+128)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(1.0 / Float64((k ^ 2.0) * t)))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(l * l) / Float64(k * k))) * Float64(cos(k) / Float64(t * Float64(0.5 - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 6.7e+128)
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / ((k ^ 2.0) * t))));
	else
		tmp = (2.0 * ((l * l) / (k * k))) * (cos(k) / (t * (0.5 - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 6.7e+128], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.69999999999999993e128

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 73.0%

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

   if 6.69999999999999993e128 < l

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 67.6%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 35.9%

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

Alternative 14: 70.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.7 \cdot 10^{+128}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 6.7e+128)
   (* 2.0 (* (/ l k) (* (/ l k) (/ 1.0 (* (pow k 2.0) t)))))
   (/ (* (* (+ l l) l) (cos k)) (* (* (* k k) t) (- 0.5 0.5)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.7e+128) {
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (pow(k, 2.0) * t))));
	} else {
		tmp = (((l + l) * l) * cos(k)) / (((k * k) * t) * (0.5 - 0.5));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 6.7d+128) then
        tmp = 2.0d0 * ((l / k) * ((l / k) * (1.0d0 / ((k ** 2.0d0) * t))))
    else
        tmp = (((l + l) * l) * cos(k)) / (((k * k) * t) * (0.5d0 - 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.7e+128) {
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (Math.pow(k, 2.0) * t))));
	} else {
		tmp = (((l + l) * l) * Math.cos(k)) / (((k * k) * t) * (0.5 - 0.5));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 6.7e+128:
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / (math.pow(k, 2.0) * t))))
	else:
		tmp = (((l + l) * l) * math.cos(k)) / (((k * k) * t) * (0.5 - 0.5))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 6.7e+128)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(1.0 / Float64((k ^ 2.0) * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) * cos(k)) / Float64(Float64(Float64(k * k) * t) * Float64(0.5 - 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 6.7e+128)
		tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / ((k ^ 2.0) * t))));
	else
		tmp = (((l + l) * l) * cos(k)) / (((k * k) * t) * (0.5 - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 6.7e+128], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.69999999999999993e128

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 85.8%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 73.0%

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

   if 6.69999999999999993e128 < l

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 74.5%

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 35.8%

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

Alternative 15: 69.8% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l k) (* (/ l k) (/ 1.0 (* (pow k 2.0) t))))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / k) * ((l / k) * (1.0 / (pow(k, 2.0) * t))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(t, l, k):
	return 2.0 * ((l / k) * ((l / k) * (1.0 / (math.pow(k, 2.0) * t))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(1.0 / Float64((k ^ 2.0) * t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / k) * ((l / k) * (1.0 / ((k ^ 2.0) * t))));
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

2. 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. Applied rewrites 74.0%

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

6. Applied rewrites 74.5%

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

8. Applied rewrites 85.8%

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

9. Taylor expanded in k around 0

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

11. Applied rewrites 73.0%

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

Alternative 16: 69.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\ell}{{k}^{4}}}{t} \cdot \left(\ell + \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot \ell}{k \cdot k}\right) \cdot \frac{1}{{k}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 3.9e-165)
   (* (/ (/ l (pow k 4.0)) t) (+ l l))
   (* (* 2.0 (/ (* l l) (* k k))) (/ 1.0 (* (pow k 2.0) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.9e-165) {
		tmp = ((l / pow(k, 4.0)) / t) * (l + l);
	} else {
		tmp = (2.0 * ((l * l) / (k * k))) * (1.0 / (pow(k, 2.0) * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.9d-165) then
        tmp = ((l / (k ** 4.0d0)) / t) * (l + l)
    else
        tmp = (2.0d0 * ((l * l) / (k * k))) * (1.0d0 / ((k ** 2.0d0) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.9e-165) {
		tmp = ((l / Math.pow(k, 4.0)) / t) * (l + l);
	} else {
		tmp = (2.0 * ((l * l) / (k * k))) * (1.0 / (Math.pow(k, 2.0) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 3.9e-165:
		tmp = ((l / math.pow(k, 4.0)) / t) * (l + l)
	else:
		tmp = (2.0 * ((l * l) / (k * k))) * (1.0 / (math.pow(k, 2.0) * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 3.9e-165)
		tmp = Float64(Float64(Float64(l / (k ^ 4.0)) / t) * Float64(l + l));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(l * l) / Float64(k * k))) * Float64(1.0 / Float64((k ^ 2.0) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 3.9e-165)
		tmp = ((l / (k ^ 4.0)) / t) * (l + l);
	else
		tmp = (2.0 * ((l * l) / (k * k))) * (1.0 / ((k ^ 2.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 3.9e-165], N[(N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.8999999999999999e-165

   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. Taylor expanded in k around 0

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

   4. Applied rewrites 63.2%

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

   6. Applied rewrites 68.9%

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

   8. Applied rewrites 68.9%

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

   10. Applied rewrites 69.6%

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

   if 3.8999999999999999e-165 < l

   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. 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. Applied rewrites 74.0%

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

   6. Applied rewrites 67.6%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 66.2%

      \[\leadsto \left(2 \cdot \frac{\ell \cdot \ell}{k \cdot k}\right) \cdot \frac{1}{\color{blue}{{k}^{2} \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 68.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\ell}{{k}^{4}}}{t} \cdot \left(\ell + \ell\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (/ l (pow k 4.0)) t) (+ l l)))

C

double code(double t, double l, double k) {
	return ((l / pow(k, 4.0)) / t) * (l + l);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / (k ** 4.0d0)) / t) * (l + l)
end function

Java

public static double code(double t, double l, double k) {
	return ((l / Math.pow(k, 4.0)) / t) * (l + l);
}

Python

def code(t, l, k):
	return ((l / math.pow(k, 4.0)) / t) * (l + l)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / (k ^ 4.0)) / t) * Float64(l + l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / (k ^ 4.0)) / t) * (l + l);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

2. Taylor expanded in k around 0

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

4. Applied rewrites 63.2%

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

6. Applied rewrites 68.9%

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

8. Applied rewrites 68.9%

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

10. Applied rewrites 69.6%

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

Alternative 18: 68.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\ell + \ell}{{k}^{4}} \cdot \frac{\ell}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (+ l l) (pow k 4.0)) (/ l t)))

C

double code(double t, double l, double k) {
	return ((l + l) / pow(k, 4.0)) * (l / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l + l) / (k ** 4.0d0)) * (l / t)
end function

Java

public static double code(double t, double l, double k) {
	return ((l + l) / Math.pow(k, 4.0)) * (l / t);
}

Python

def code(t, l, k):
	return ((l + l) / math.pow(k, 4.0)) * (l / t)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l + l) / (k ^ 4.0)) * Float64(l / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l + l) / (k ^ 4.0)) * (l / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l + l), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

2. Taylor expanded in k around 0

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

4. Applied rewrites 63.2%

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

6. Applied rewrites 68.9%

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

8. Applied rewrites 68.9%

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

10. Applied rewrites 68.1%

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

Alternative 19: 66.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (pow k 4.0) t)) (+ l l)))

C

double code(double t, double l, double k) {
	return (l / (pow(k, 4.0) * t)) * (l + l);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / ((k ** 4.0d0) * t)) * (l + l)
end function

Java

public static double code(double t, double l, double k) {
	return (l / (Math.pow(k, 4.0) * t)) * (l + l);
}

Python

def code(t, l, k):
	return (l / (math.pow(k, 4.0) * t)) * (l + l)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64((k ^ 4.0) * t)) * Float64(l + l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k ^ 4.0) * t)) * (l + l);
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

2. Taylor expanded in k around 0

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

4. Applied rewrites 63.2%

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

6. Applied rewrites 68.9%

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

8. Applied rewrites 68.9%

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

Reproduce

herbie shell --seed 2025153 
(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))))