Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 93.7%

Time: 8.6s

Alternatives: 13

Speedup: 4.9×

• 39.2% 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: 93.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell + \ell}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-6)
   (* (/ (/ (+ l l) k_m) k_m) (/ l (* k_m (* k_m t))))
   (*
    (* (/ (+ l l) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t)) (/ l k_m))
    (/ (cos k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (((l + l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (cos(k_m) / 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 <= 2.4d-6) then
        tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
    else
        tmp = (((l + l) / ((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t)) * (l / k_m)) * (cos(k_m) / 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 <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (((l + l) / ((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (Math.cos(k_m) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-6:
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
	else:
		tmp = (((l + l) / ((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (math.cos(k_m) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t)) * Float64(l / k_m)) * Float64(cos(k_m) / 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 <= 2.4e-6)
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	else
		tmp = (((l + l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (cos(k_m) / 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, 2.4e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3999999999999999e-6

   1. Initial program 41.1%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 2.3999999999999999e-6 < k

   1. Initial program 30.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sin.f64 81.0

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

   8. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   10. Applied rewrites 91.4%

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

Alternative 2: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5} \cdot \frac{\ell + \ell}{k\_m \cdot t}\right) \cdot \frac{\cos k\_m}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-6)
   (* (/ (/ (+ l l) k_m) k_m) (/ l (* k_m (* k_m t))))
   (*
    (* (/ l (- 0.5 (* (cos (+ k_m k_m)) 0.5))) (/ (+ l l) (* k_m t)))
    (/ (cos k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = ((l / (0.5 - (cos((k_m + k_m)) * 0.5))) * ((l + l) / (k_m * t))) * (cos(k_m) / 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 <= 2.4d-6) then
        tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
    else
        tmp = ((l / (0.5d0 - (cos((k_m + k_m)) * 0.5d0))) * ((l + l) / (k_m * t))) * (cos(k_m) / 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 <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = ((l / (0.5 - (Math.cos((k_m + k_m)) * 0.5))) * ((l + l) / (k_m * t))) * (Math.cos(k_m) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-6:
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
	else:
		tmp = ((l / (0.5 - (math.cos((k_m + k_m)) * 0.5))) * ((l + l) / (k_m * t))) * (math.cos(k_m) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(Float64(l / Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5))) * Float64(Float64(l + l) / Float64(k_m * t))) * Float64(cos(k_m) / 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 <= 2.4e-6)
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	else
		tmp = ((l / (0.5 - (cos((k_m + k_m)) * 0.5))) * ((l + l) / (k_m * t))) * (cos(k_m) / 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, 2.4e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3999999999999999e-6

   1. Initial program 41.1%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 2.3999999999999999e-6 < k

   1. Initial program 30.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 80.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sin.f64 81.0

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

   8. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   10. Applied rewrites 88.0%

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

Alternative 3: 89.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := 0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\\ \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{elif}\;k\_m \leq 4.6 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\ell + \ell\right) \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\right) \cdot \frac{\cos k\_m}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \frac{\cos k\_m}{k\_m}}{\left(t\_1 \cdot t\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (- 0.5 (* (cos (+ k_m k_m)) 0.5))))
   (if (<= k_m 2.4e-6)
     (* (/ (/ (+ l l) k_m) k_m) (/ l (* k_m (* k_m t))))
     (if (<= k_m 4.6e+138)
       (* (* (+ l l) (/ l (* (* k_m k_m) t))) (/ (cos k_m) t_1))
       (/ (* (* (+ l l) l) (/ (cos k_m) k_m)) (* (* t_1 t) k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 0.5 - (cos((k_m + k_m)) * 0.5);
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else if (k_m <= 4.6e+138) {
		tmp = ((l + l) * (l / ((k_m * k_m) * t))) * (cos(k_m) / t_1);
	} else {
		tmp = (((l + l) * l) * (cos(k_m) / k_m)) / ((t_1 * t) * 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) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 - (cos((k_m + k_m)) * 0.5d0)
    if (k_m <= 2.4d-6) then
        tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
    else if (k_m <= 4.6d+138) then
        tmp = ((l + l) * (l / ((k_m * k_m) * t))) * (cos(k_m) / t_1)
    else
        tmp = (((l + l) * l) * (cos(k_m) / k_m)) / ((t_1 * t) * 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 t_1 = 0.5 - (Math.cos((k_m + k_m)) * 0.5);
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else if (k_m <= 4.6e+138) {
		tmp = ((l + l) * (l / ((k_m * k_m) * t))) * (Math.cos(k_m) / t_1);
	} else {
		tmp = (((l + l) * l) * (Math.cos(k_m) / k_m)) / ((t_1 * t) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 0.5 - (math.cos((k_m + k_m)) * 0.5)
	tmp = 0
	if k_m <= 2.4e-6:
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
	elif k_m <= 4.6e+138:
		tmp = ((l + l) * (l / ((k_m * k_m) * t))) * (math.cos(k_m) / t_1)
	else:
		tmp = (((l + l) * l) * (math.cos(k_m) / k_m)) / ((t_1 * t) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5))
	tmp = 0.0
	if (k_m <= 2.4e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(k_m * t))));
	elseif (k_m <= 4.6e+138)
		tmp = Float64(Float64(Float64(l + l) * Float64(l / Float64(Float64(k_m * k_m) * t))) * Float64(cos(k_m) / t_1));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) * Float64(cos(k_m) / k_m)) / Float64(Float64(t_1 * t) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 0.5 - (cos((k_m + k_m)) * 0.5);
	tmp = 0.0;
	if (k_m <= 2.4e-6)
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	elseif (k_m <= 4.6e+138)
		tmp = ((l + l) * (l / ((k_m * k_m) * t))) * (cos(k_m) / t_1);
	else
		tmp = (((l + l) * l) * (cos(k_m) / k_m)) / ((t_1 * t) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2.4e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.6e+138], N[(N[(N[(l + l), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.3999999999999999e-6

   1. Initial program 41.1%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 2.3999999999999999e-6 < k < 4.60000000000000015e138

   1. Initial program 22.7%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 82.6%

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

   6. Applied rewrites 83.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sin.f64 83.9

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

   8. Applied rewrites 83.9%

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

   10. Applied rewrites 90.4%

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

   if 4.60000000000000015e138 < k

   1. Initial program 37.2%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 61.8%

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

   6. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-*l/ N/A

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

   8. Applied rewrites 78.4%

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

Alternative 4: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m} \cdot \frac{\cos k\_m}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-6)
   (* (/ (/ (+ l l) k_m) k_m) (/ l (* k_m (* k_m t))))
   (*
    (/ (* (+ l l) l) (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m))
    (/ (cos k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (((l + l) * l) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m)) * (cos(k_m) / 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 <= 2.4d-6) then
        tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
    else
        tmp = (((l + l) * l) / (((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m)) * (cos(k_m) / 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 <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (((l + l) * l) / (((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m)) * (Math.cos(k_m) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-6:
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
	else:
		tmp = (((l + l) * l) / (((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m)) * (math.cos(k_m) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m)) * Float64(cos(k_m) / 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 <= 2.4e-6)
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	else
		tmp = (((l + l) * l) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m)) * (cos(k_m) / 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, 2.4e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3999999999999999e-6

   1. Initial program 41.1%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 2.3999999999999999e-6 < k

   1. Initial program 30.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 80.5

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

   8. Applied rewrites 80.5%

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

Alternative 5: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-6)
   (* (/ (/ (+ l l) k_m) k_m) (/ l (* k_m (* k_m t))))
   (/
    (* (* (* (cos k_m) l) l) 2.0)
    (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * 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 <= 2.4d-6) then
        tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
    else
        tmp = (((cos(k_m) * l) * l) * 2.0d0) / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m) * 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 <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (((Math.cos(k_m) * l) * l) * 2.0) / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-6:
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
	else:
		tmp = (((math.cos(k_m) * l) * l) * 2.0) / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * 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 <= 2.4e-6)
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	else
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * 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, 2.4e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3999999999999999e-6

   1. Initial program 41.1%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 2.3999999999999999e-6 < k

   1. Initial program 30.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 77.3%

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

Alternative 6: 86.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \left(\left(\ell + \ell\right) \cdot \ell\right)}{k\_m \cdot \left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-6)
   (* (/ (/ (+ l l) k_m) k_m) (/ l (* k_m (* k_m t))))
   (/
    (* (cos k_m) (* (+ l l) l))
    (* k_m (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (cos(k_m) * ((l + l) * l)) / (k_m * (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * 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 <= 2.4d-6) then
        tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
    else
        tmp = (cos(k_m) * ((l + l) * l)) / (k_m * (((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * 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 <= 2.4e-6) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (Math.cos(k_m) * ((l + l) * l)) / (k_m * (((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-6:
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
	else:
		tmp = (math.cos(k_m) * ((l + l) * l)) / (k_m * (((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(cos(k_m) * Float64(Float64(l + l) * l)) / Float64(k_m * Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * 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 <= 2.4e-6)
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	else
		tmp = (cos(k_m) * ((l + l) * l)) / (k_m * (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * 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, 2.4e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3999999999999999e-6

   1. Initial program 41.1%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 93.5

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

   6. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 96.2

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

   10. Applied rewrites 96.2%

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

   if 2.3999999999999999e-6 < k

   1. Initial program 30.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 71.3%

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

   6. Applied rewrites 80.5%

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

   8. Applied rewrites 77.2%

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

Alternative 7: 76.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \frac{\ell \cdot -0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (* (/ (/ (+ l l) k_m) k_m) (/ l (* k_m (* k_m t))))
   (* (/ l k_m) (/ (* l -0.3333333333333333) (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	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 <= 26.5d0) then
        tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
    else
        tmp = (l / k_m) * ((l * (-0.3333333333333333d0)) / (k_m * t))
    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 <= 26.5) {
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 26.5:
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)))
	else:
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(Float64(Float64(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l * -0.3333333333333333) / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 26.5)
		tmp = (((l + l) / k_m) / k_m) * (l / (k_m * (k_m * t)));
	else
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 40.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 92.8

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 92.8

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

   8. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 95.4

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

   10. Applied rewrites 95.4%

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

   if 26.5 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 55.0

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.0%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lift-*.f64 57.9

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 57.9%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \frac{\ell \cdot -0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* k_m (* k_m t))))
   (* (/ l k_m) (/ (* l -0.3333333333333333) (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	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 <= 26.5d0) then
        tmp = ((l + l) / (k_m * k_m)) * (l / (k_m * (k_m * t)))
    else
        tmp = (l / k_m) * ((l * (-0.3333333333333333d0)) / (k_m * t))
    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 <= 26.5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / (k_m * (k_m * t)));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 26.5:
		tmp = ((l + l) / (k_m * k_m)) * (l / (k_m * (k_m * t)))
	else:
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l * -0.3333333333333333) / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 26.5)
		tmp = ((l + l) / (k_m * k_m)) * (l / (k_m * (k_m * t)));
	else
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 40.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 92.8

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 92.8

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

   8. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 92.8

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

   10. Applied rewrites 92.8%

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

   if 26.5 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 55.0

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.0%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lift-*.f64 57.9

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 57.9%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \frac{\ell \cdot -0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (/ (* 2.0 (* l l)) (* (* k_m k_m) (* k_m (* k_m t))))
   (* (/ l k_m) (/ (* l -0.3333333333333333) (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	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 <= 26.5d0) then
        tmp = (2.0d0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)))
    else
        tmp = (l / k_m) * ((l * (-0.3333333333333333d0)) / (k_m * t))
    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 <= 26.5) {
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 26.5:
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)))
	else:
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(k_m * k_m) * Float64(k_m * Float64(k_m * t))));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l * -0.3333333333333333) / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 26.5)
		tmp = (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)));
	else
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 40.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 76.1

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

   6. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 76.1

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

   8. Applied rewrites 76.1%

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

   if 26.5 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 55.0

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.0%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lift-*.f64 57.9

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 57.9%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 26.5:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \frac{\ell \cdot -0.3333333333333333}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 26.5)
   (/ (* (+ l l) l) (* (* (* k_m k_m) t) (* k_m k_m)))
   (* (/ l k_m) (/ (* l -0.3333333333333333) (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 26.5) {
		tmp = ((l + l) * l) / (((k_m * k_m) * t) * (k_m * k_m));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	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 <= 26.5d0) then
        tmp = ((l + l) * l) / (((k_m * k_m) * t) * (k_m * k_m))
    else
        tmp = (l / k_m) * ((l * (-0.3333333333333333d0)) / (k_m * t))
    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 <= 26.5) {
		tmp = ((l + l) * l) / (((k_m * k_m) * t) * (k_m * k_m));
	} else {
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 26.5:
		tmp = ((l + l) * l) / (((k_m * k_m) * t) * (k_m * k_m))
	else:
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 26.5)
		tmp = Float64(Float64(Float64(l + l) * l) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l * -0.3333333333333333) / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 26.5)
		tmp = ((l + l) * l) / (((k_m * k_m) * t) * (k_m * k_m));
	else
		tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 26.5], N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26.5

   1. Initial program 40.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 92.8

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

   6. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r* N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. pow2 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   8. Applied rewrites 76.1%

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

   if 26.5 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 55.0

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.0%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lift-*.f64 57.9

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 57.9%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{k\_m} \cdot \frac{\ell \cdot -0.3333333333333333}{k\_m \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l k_m) (/ (* l -0.3333333333333333) (* k_m t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / k_m) * ((l * -0.3333333333333333) / (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) * ((l * (-0.3333333333333333d0)) / (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) * ((l * -0.3333333333333333) / (k_m * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / k_m) * ((l * -0.3333333333333333) / (k_m * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / k_m) * Float64(Float64(l * -0.3333333333333333) / Float64(k_m * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / k_m) * ((l * -0.3333333333333333) / (k_m * t));
end

Wolfram

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

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.7%

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

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 29.7

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.7%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

   13. lift-*.f64 31.2

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

9. Applied rewrites 31.2%

   \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
10. Add Preprocessing

Alternative 12: 30.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k\_m \cdot t\right) \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (* l l) -0.3333333333333333) (* (* k_m t) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / ((k_m * t) * k_m);
}

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 * l) * (-0.3333333333333333d0)) / ((k_m * t) * k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / ((k_m * t) * k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) * -0.3333333333333333) / ((k_m * t) * k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(Float64(k_m * t) * k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) * -0.3333333333333333) / ((k_m * t) * k_m);
end

Wolfram

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

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.7%

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

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 29.7

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.7%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 30.2

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot t\right) \cdot k} \]

9. Applied rewrites 30.2%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot t\right) \cdot k} \]
10. Add Preprocessing

Alternative 13: 29.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot -0.3333333333333333 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* l l) (* (* k_m k_m) t)) -0.3333333333333333))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
}

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 * l) / ((k_m * k_m) * t)) * (-0.3333333333333333d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * t)) * -0.3333333333333333)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
end

Wolfram

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

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.7%

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

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 29.7

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.7%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 29.7

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

9. Applied rewrites 29.7%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(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))))