Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 97.5%

Time: 8.0s

Alternatives: 9

Speedup: 11.6×

• 40.3% 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: 35.7% 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: 97.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot 2\\ \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k\_m}}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k\_m}}{{\sin k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) 2.0)))
   (if (<= k_m 9.2e-169)
     (* (* (/ (/ l k_m) (* (* k_m t) k_m)) (/ l k_m)) t_1)
     (* (* (/ (/ l k_m) (* (pow (sin k_m) 2.0) t)) (/ l k_m)) t_1))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * 2.0;
	double tmp;
	if (k_m <= 9.2e-169) {
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	} else {
		tmp = (((l / k_m) / (pow(sin(k_m), 2.0) * t)) * (l / k_m)) * t_1;
	}
	return tmp;
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * 2.0d0
    if (k_m <= 9.2d-169) then
        tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1
    else
        tmp = (((l / k_m) / ((sin(k_m) ** 2.0d0) * t)) * (l / k_m)) * t_1
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) * 2.0;
	double tmp;
	if (k_m <= 9.2e-169) {
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	} else {
		tmp = (((l / k_m) / (Math.pow(Math.sin(k_m), 2.0) * t)) * (l / k_m)) * t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * 2.0
	tmp = 0
	if k_m <= 9.2e-169:
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1
	else:
		tmp = (((l / k_m) / (math.pow(math.sin(k_m), 2.0) * t)) * (l / k_m)) * t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * 2.0)
	tmp = 0.0
	if (k_m <= 9.2e-169)
		tmp = Float64(Float64(Float64(Float64(l / k_m) / Float64(Float64(k_m * t) * k_m)) * Float64(l / k_m)) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(l / k_m) / Float64((sin(k_m) ^ 2.0) * t)) * Float64(l / k_m)) * t_1);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * 2.0;
	tmp = 0.0;
	if (k_m <= 9.2e-169)
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	else
		tmp = (((l / k_m) / ((sin(k_m) ^ 2.0) * t)) * (l / k_m)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[k$95$m, 9.2e-169], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.2000000000000004e-169

   1. Initial program 38.4%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 81.3%

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

   7. Applied rewrites 92.1%

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

   9. Applied rewrites 94.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.8%

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

   if 9.2000000000000004e-169 < k

   1. Initial program 27.3%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 90.8%

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

   9. Applied rewrites 98.7%

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

Alternative 2: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot 2\\ \mathbf{if}\;k\_m \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k\_m}}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) 2.0)))
   (if (<= k_m 5.8e-45)
     (* (* (/ (/ l k_m) (* (* k_m t) k_m)) (/ l k_m)) t_1)
     (* (* (/ l (* (* (pow (sin k_m) 2.0) t) k_m)) (/ l k_m)) t_1))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * 2.0;
	double tmp;
	if (k_m <= 5.8e-45) {
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	} else {
		tmp = ((l / ((pow(sin(k_m), 2.0) * t) * k_m)) * (l / k_m)) * t_1;
	}
	return tmp;
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * 2.0d0
    if (k_m <= 5.8d-45) then
        tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1
    else
        tmp = ((l / (((sin(k_m) ** 2.0d0) * t) * k_m)) * (l / k_m)) * t_1
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) * 2.0;
	double tmp;
	if (k_m <= 5.8e-45) {
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	} else {
		tmp = ((l / ((Math.pow(Math.sin(k_m), 2.0) * t) * k_m)) * (l / k_m)) * t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * 2.0
	tmp = 0
	if k_m <= 5.8e-45:
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1
	else:
		tmp = ((l / ((math.pow(math.sin(k_m), 2.0) * t) * k_m)) * (l / k_m)) * t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * 2.0)
	tmp = 0.0
	if (k_m <= 5.8e-45)
		tmp = Float64(Float64(Float64(Float64(l / k_m) / Float64(Float64(k_m * t) * k_m)) * Float64(l / k_m)) * t_1);
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m)) * Float64(l / k_m)) * t_1);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * 2.0;
	tmp = 0.0;
	if (k_m <= 5.8e-45)
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	else
		tmp = ((l / (((sin(k_m) ^ 2.0) * t) * k_m)) * (l / k_m)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[k$95$m, 5.8e-45], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(l / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e-45

   1. Initial program 36.6%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 91.1%

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

   9. Applied rewrites 94.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 82.8%

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

   if 5.8e-45 < k

   1. Initial program 29.8%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 93.1%

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

Alternative 3: 94.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot 2\\ \mathbf{if}\;k\_m \leq 8.4 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\frac{\ell}{k\_m}}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) 2.0)))
   (if (<= k_m 8.4e-5)
     (* (* (/ (/ l k_m) (* (* k_m t) k_m)) (/ l k_m)) t_1)
     (*
      (* (/ l (* (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t) k_m)) (/ l k_m))
      t_1))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * 2.0;
	double tmp;
	if (k_m <= 8.4e-5) {
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	} else {
		tmp = ((l / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * k_m)) * (l / k_m)) * t_1;
	}
	return tmp;
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * 2.0d0
    if (k_m <= 8.4d-5) then
        tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1
    else
        tmp = ((l / (((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t) * k_m)) * (l / k_m)) * t_1
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) * 2.0;
	double tmp;
	if (k_m <= 8.4e-5) {
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	} else {
		tmp = ((l / (((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t) * k_m)) * (l / k_m)) * t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * 2.0
	tmp = 0
	if k_m <= 8.4e-5:
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1
	else:
		tmp = ((l / (((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t) * k_m)) * (l / k_m)) * t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * 2.0)
	tmp = 0.0
	if (k_m <= 8.4e-5)
		tmp = Float64(Float64(Float64(Float64(l / k_m) / Float64(Float64(k_m * t) * k_m)) * Float64(l / k_m)) * t_1);
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t) * k_m)) * Float64(l / k_m)) * t_1);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * 2.0;
	tmp = 0.0;
	if (k_m <= 8.4e-5)
		tmp = (((l / k_m) / ((k_m * t) * k_m)) * (l / k_m)) * t_1;
	else
		tmp = ((l / (((0.5 - (0.5 * cos((2.0 * k_m)))) * t) * k_m)) * (l / k_m)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[k$95$m, 8.4e-5], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(l / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.39999999999999954e-5

   1. Initial program 36.4%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 94.9%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 83.2%

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

   if 8.39999999999999954e-5 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 92.4%

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

   9. Applied rewrites 92.1%

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

Alternative 4: 76.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 11.5:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\ell \cdot \ell}{t}}{k\_m} \cdot \frac{0.3333333333333333}{k\_m}\right) \cdot \left(2 \cdot \cos k\_m\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 11.5)
   (* (/ (/ l (* k_m k_m)) (* (* k_m k_m) t)) (* 2.0 l))
   (* (* (/ (/ (* l l) t) k_m) (/ 0.3333333333333333 k_m)) (* 2.0 (cos k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 11.5) {
		tmp = ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l);
	} else {
		tmp = ((((l * l) / t) / k_m) * (0.3333333333333333 / k_m)) * (2.0 * cos(k_m));
	}
	return tmp;
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 11.5d0) then
        tmp = ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0d0 * l)
    else
        tmp = ((((l * l) / t) / k_m) * (0.3333333333333333d0 / k_m)) * (2.0d0 * cos(k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 11.5) {
		tmp = ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l);
	} else {
		tmp = ((((l * l) / t) / k_m) * (0.3333333333333333 / k_m)) * (2.0 * Math.cos(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 11.5:
		tmp = ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l)
	else:
		tmp = ((((l * l) / t) / k_m) * (0.3333333333333333 / k_m)) * (2.0 * math.cos(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 11.5)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / Float64(Float64(k_m * k_m) * t)) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / t) / k_m) * Float64(0.3333333333333333 / k_m)) * Float64(2.0 * cos(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 11.5)
		tmp = ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l);
	else
		tmp = ((((l * l) / t) / k_m) * (0.3333333333333333 / k_m)) * (2.0 * cos(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 11.5], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(0.3333333333333333 / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 11.5

   1. Initial program 36.4%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 75.6%

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

   9. Applied rewrites 78.4%

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

   if 11.5 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 18.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 59.5%

       \[\leadsto \left(\frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \frac{0.3333333333333333}{k}\right) \cdot \left(2 \cdot \cos k\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.6% accurate, 2.9× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* (/ (/ l k_m) (* (* k_m t) k_m)) (/ l k_m)) (* (cos k_m) 2.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(l / k_m) / Float64(Float64(k_m * t) * k_m)) * Float64(l / k_m)) * Float64(cos(k_m) * 2.0))
end

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in t around 0

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

   1. count-2-rev N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 80.2%

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

7. Applied rewrites 91.7%

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

9. Applied rewrites 96.0%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 77.1%

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

Alternative 6: 72.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\ell}{k\_m \cdot k\_m}}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(2 \cdot \ell\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (/ l (* k_m k_m)) (* (* k_m k_m) t)) (* 2.0 l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l);
}

Fortran

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

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l / Float64(k_m * k_m)) / Float64(Float64(k_m * k_m) * t)) * Float64(2.0 * l))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l / (k_m * k_m)) / ((k_m * k_m) * t)) * (2.0 * l);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.6

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

5. Applied rewrites 70.6%

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

7. Applied rewrites 70.5%

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

9. Applied rewrites 72.7%

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

Alternative 7: 72.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* k_m k_m)) (/ (* l 2.0) (* (* k_m k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (k_m * k_m)) * ((l * 2.0) / ((k_m * k_m) * t));
}

Fortran

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

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / (k_m * k_m)) * ((l * 2.0) / ((k_m * k_m) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / (k_m * k_m)) * ((l * 2.0) / ((k_m * k_m) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * 2.0) / Float64(Float64(k_m * k_m) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / (k_m * k_m)) * ((l * 2.0) / ((k_m * k_m) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.6

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

5. Applied rewrites 70.6%

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

7. Applied rewrites 70.5%

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

9. Applied rewrites 72.7%

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

Alternative 8: 70.5% accurate, 11.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{k\_m \cdot \left(k\_m \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)\right)} \cdot \left(2 \cdot \ell\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* k_m (* k_m (* (* k_m k_m) t)))) (* 2.0 l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (k_m * (k_m * ((k_m * k_m) * t)))) * (2.0 * l);
}

Fortran

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

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / (k_m * (k_m * ((k_m * k_m) * t)))) * (2.0 * l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / (k_m * (k_m * ((k_m * k_m) * t)))) * (2.0 * l)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64(k_m * Float64(k_m * Float64(Float64(k_m * k_m) * t)))) * Float64(2.0 * l))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / (k_m * (k_m * ((k_m * k_m) * t)))) * (2.0 * l);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.6

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

5. Applied rewrites 70.6%

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

7. Applied rewrites 70.5%

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

9. Applied rewrites 71.4%

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

Alternative 9: 70.5% accurate, 11.6× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* (* k_m k_m) (* (* k_m k_m) t))) (+ l l)))

C

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

Fortran

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

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / ((k_m * k_m) * ((k_m * k_m) * t))) * (l + l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / ((k_m * k_m) * ((k_m * k_m) * t))) * (l + l)

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.6

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

5. Applied rewrites 70.6%

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

7. Applied rewrites 70.5%

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

9. Applied rewrites 70.5%

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

11. Applied rewrites 71.4%

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

Reproduce

herbie shell --seed 2025017 
(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))))