Toniolo and Linder, Equation (10-)

Percentage Accurate: 37.2% → 98.7%

Time: 11.6s

Alternatives: 20

Speedup: 8.6×

• 40.1% 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: 37.2% 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: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 1.1 \cdot 10^{-144}:\\ \;\;\;\;\left(\frac{2}{k\_m} \cdot t\_1\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot 2}{k\_m} \cdot \frac{\frac{\frac{\ell}{{\sin k\_m}^{2}}}{k\_m}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 1.1e-144)
     (*
      (* (/ 2.0 k_m) t_1)
      (/
       (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m)
       (* t k_m)))
     (* (/ (* t_1 2.0) k_m) (/ (/ (/ l (pow (sin k_m) 2.0)) k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 1.1e-144) {
		tmp = ((2.0 / k_m) * t_1) * (((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m) / (t * k_m));
	} else {
		tmp = ((t_1 * 2.0) / k_m) * (((l / pow(sin(k_m), 2.0)) / k_m) / t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 1.1e-144)
		tmp = Float64(Float64(Float64(2.0 / k_m) * t_1) * Float64(Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m) / Float64(t * k_m)));
	else
		tmp = Float64(Float64(Float64(t_1 * 2.0) / k_m) * Float64(Float64(Float64(l / (sin(k_m) ^ 2.0)) / k_m) / t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.1e-144], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000003e-144

   1. Initial program 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.3

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 83.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 70.0%

       \[\leadsto \frac{\frac{\left(2 \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k}}{k}}{k \cdot k}}{t} \]
   13. Step-by-step derivation

   14. Applied rewrites 78.0%

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

   if 1.10000000000000003e-144 < k

   1. Initial program 32.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 37.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 37.2

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

   3. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 66.3%

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

   9. Applied rewrites 72.4%

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

   11. Applied rewrites 98.6%

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

Alternative 2: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.1 \cdot 10^{-144}:\\ \;\;\;\;\left(\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\frac{\frac{\ell}{{\sin k\_m}^{2}}}{k\_m}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.1e-144)
   (*
    (* (/ 2.0 k_m) (* (cos k_m) l))
    (/ (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m) (* t k_m)))
   (*
    (* (* (cos k_m) 2.0) (/ l k_m))
    (/ (/ (/ l (pow (sin k_m) 2.0)) k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e-144) {
		tmp = ((2.0 / k_m) * (cos(k_m) * l)) * (((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m) / (t * k_m));
	} else {
		tmp = ((cos(k_m) * 2.0) * (l / k_m)) * (((l / pow(sin(k_m), 2.0)) / k_m) / t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e-144)
		tmp = Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) * Float64(Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m) / Float64(t * k_m)));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / k_m)) * Float64(Float64(Float64(l / (sin(k_m) ^ 2.0)) / k_m) / t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.1e-144], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000003e-144

   1. Initial program 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.3

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 83.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 70.0%

       \[\leadsto \frac{\frac{\left(2 \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k}}{k}}{k \cdot k}}{t} \]
   13. Step-by-step derivation

   14. Applied rewrites 78.0%

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

   if 1.10000000000000003e-144 < k

   1. Initial program 32.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 37.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 37.2

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

   3. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 66.3%

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

   9. Applied rewrites 72.4%

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

   11. Applied rewrites 98.6%

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

   13. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 0.0025:\\ \;\;\;\;\left(\frac{2}{k\_m} \cdot t\_1\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot 2}{k\_m} \cdot \frac{\frac{\frac{\ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}}{k\_m}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 0.0025)
     (*
      (* (/ 2.0 k_m) t_1)
      (/
       (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m)
       (* t k_m)))
     (*
      (/ (* t_1 2.0) k_m)
      (/ (/ (/ l (- 0.5 (* 0.5 (cos (* 2.0 k_m))))) k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 0.0025) {
		tmp = ((2.0 / k_m) * t_1) * (((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m) / (t * k_m));
	} else {
		tmp = ((t_1 * 2.0) / k_m) * (((l / (0.5 - (0.5 * cos((2.0 * k_m))))) / k_m) / t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 0.0025)
		tmp = Float64(Float64(Float64(2.0 / k_m) * t_1) * Float64(Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m) / Float64(t * k_m)));
	else
		tmp = Float64(Float64(Float64(t_1 * 2.0) / k_m) * Float64(Float64(Float64(l / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))) / k_m) / t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0025], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 35.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.4

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

   3. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 74.8%

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

   9. Applied rewrites 84.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 73.3%

       \[\leadsto \frac{\frac{\left(2 \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k}}{k}}{k \cdot k}}{t} \]
   13. Step-by-step derivation

   14. Applied rewrites 81.3%

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

   if 0.00250000000000000005 < k

   1. Initial program 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 35.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 35.6

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

   3. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 63.3%

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

   11. Applied rewrites 99.5%

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

   13. Applied rewrites 99.0%

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

Alternative 4: 86.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}}{t \cdot k\_m}\\ \mathbf{elif}\;k\_m \leq 2.55 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(\frac{\frac{\sin k\_m}{\ell}}{\ell} \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k\_m} \cdot \frac{\frac{\frac{\ell}{{\sin k\_m}^{2}}}{k\_m}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.4e-16)
   (*
    (* (/ 2.0 k_m) (* (cos k_m) l))
    (/ (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m) (* t k_m)))
   (if (<= k_m 2.55e+144)
     (/ 2.0 (* (* (* k_m k_m) t) (* (/ (/ (sin k_m) l) l) (tan k_m))))
     (* (/ (* l 2.0) k_m) (/ (/ (/ l (pow (sin k_m) 2.0)) k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e-16) {
		tmp = ((2.0 / k_m) * (cos(k_m) * l)) * (((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m) / (t * k_m));
	} else if (k_m <= 2.55e+144) {
		tmp = 2.0 / (((k_m * k_m) * t) * (((sin(k_m) / l) / l) * tan(k_m)));
	} else {
		tmp = ((l * 2.0) / k_m) * (((l / pow(sin(k_m), 2.0)) / k_m) / t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.4e-16)
		tmp = Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) * Float64(Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m) / Float64(t * k_m)));
	elseif (k_m <= 2.55e+144)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(Float64(sin(k_m) / l) / l) * tan(k_m))));
	else
		tmp = Float64(Float64(Float64(l * 2.0) / k_m) * Float64(Float64(Float64(l / (sin(k_m) ^ 2.0)) / k_m) / t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.4e-16], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.55e+144], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * 2.0), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.4e-16

   1. Initial program 36.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.7

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

   3. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 74.1%

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

   9. Applied rewrites 83.6%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 72.6%

       \[\leadsto \frac{\frac{\left(2 \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k}}{k}}{k \cdot k}}{t} \]
   13. Step-by-step derivation

   14. Applied rewrites 80.8%

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

   if 3.4e-16 < k < 2.5499999999999999e144

   1. Initial program 24.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.0%

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

   if 2.5499999999999999e144 < k

   1. Initial program 32.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 38.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 38.1

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

   3. Applied rewrites 40.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 47.3%

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

   9. Applied rewrites 50.1%

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

   11. Applied rewrites 99.5%

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

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 58.2%

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

Alternative 5: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 0.0025:\\ \;\;\;\;\left(\frac{2}{k\_m} \cdot t\_1\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \ell\right) \cdot t\_1}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 0.0025)
     (*
      (* (/ 2.0 k_m) t_1)
      (/
       (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m)
       (* t k_m)))
     (/
      (* (* (/ 2.0 (* (* k_m k_m) t)) l) t_1)
      (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 0.0025) {
		tmp = ((2.0 / k_m) * t_1) * (((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m) / (t * k_m));
	} else {
		tmp = (((2.0 / ((k_m * k_m) * t)) * l) * t_1) / (0.5 - (0.5 * cos((2.0 * k_m))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 0.0025)
		tmp = Float64(Float64(Float64(2.0 / k_m) * t_1) * Float64(Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m) / Float64(t * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * l) * t_1) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0025], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 35.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.4

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

   3. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 74.8%

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

   9. Applied rewrites 84.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 73.3%

       \[\leadsto \frac{\frac{\left(2 \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k}}{k}}{k \cdot k}}{t} \]
   13. Step-by-step derivation

   14. Applied rewrites 81.3%

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

   if 0.00250000000000000005 < k

   1. Initial program 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 35.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 35.6

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

   3. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 65.8%

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

   11. Applied rewrites 65.7%

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

Alternative 6: 85.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(\frac{\frac{\sin k\_m}{\ell}}{\ell} \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.4e-16)
   (*
    (* (/ 2.0 k_m) (* (cos k_m) l))
    (/ (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m) (* t k_m)))
   (/ 2.0 (* (* (* k_m k_m) t) (* (/ (/ (sin k_m) l) l) (tan k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e-16) {
		tmp = ((2.0 / k_m) * (cos(k_m) * l)) * (((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m) / (t * k_m));
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * (((sin(k_m) / l) / l) * tan(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.4e-16)
		tmp = Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) * Float64(Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m) / Float64(t * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(Float64(sin(k_m) / l) / l) * tan(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.4e-16], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4e-16

   1. Initial program 36.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.7

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

   3. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 74.1%

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

   9. Applied rewrites 83.6%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 72.6%

       \[\leadsto \frac{\frac{\left(2 \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k}}{k}}{k \cdot k}}{t} \]
   13. Step-by-step derivation

   14. Applied rewrites 80.8%

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

   if 3.4e-16 < k

   1. Initial program 28.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 61.8%

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

Alternative 7: 79.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.35 \cdot 10^{+112}:\\ \;\;\;\;\left(\frac{2}{k\_m} \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot \ell, \ell\right)}{k\_m}}{k\_m}}{t \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k\_m} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.35e+112)
   (*
    (* (/ 2.0 k_m) (* (cos k_m) l))
    (/ (/ (/ (fma 0.3333333333333333 (* (* k_m k_m) l) l) k_m) k_m) (* t k_m)))
   (* (/ -0.3333333333333333 k_m) (* (/ l t) (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.35e+112) {
		tmp = ((2.0 / k_m) * (cos(k_m) * l)) * (((fma(0.3333333333333333, ((k_m * k_m) * l), l) / k_m) / k_m) / (t * k_m));
	} else {
		tmp = (-0.3333333333333333 / k_m) * ((l / t) * (l / k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.35e+112)
		tmp = Float64(Float64(Float64(2.0 / k_m) * Float64(cos(k_m) * l)) * Float64(Float64(Float64(fma(0.3333333333333333, Float64(Float64(k_m * k_m) * l), l) / k_m) / k_m) / Float64(t * k_m)));
	else
		tmp = Float64(Float64(-0.3333333333333333 / k_m) * Float64(Float64(l / t) * Float64(l / k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.35e+112], N[(N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / k$95$m), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.34999999999999999e112

   1. Initial program 34.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.0

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

   3. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 84.2%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 71.0%

       \[\leadsto \frac{\frac{\left(2 \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, 0.3333333333333333, \ell\right)}{k}}{k}}{k \cdot k}}{t} \]
   13. Step-by-step derivation

   14. Applied rewrites 78.4%

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

   if 2.34999999999999999e112 < k

   1. Initial program 31.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 36.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 36.1

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

   3. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 47.9%

       \[\leadsto \frac{-0.3333333333333333}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.1%

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

Alternative 8: 77.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 0.4:\\ \;\;\;\;\frac{t\_1 \cdot 2}{k\_m} \cdot \frac{\frac{\frac{\frac{\ell}{k\_m}}{k\_m}}{k\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot t\_1\right) \cdot \left(0.3333333333333333 \cdot \ell\right)}{k\_m \cdot k\_m}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 0.4)
     (* (/ (* t_1 2.0) k_m) (/ (/ (/ (/ l k_m) k_m) k_m) t))
     (/ (/ (* (* 2.0 t_1) (* 0.3333333333333333 l)) (* k_m k_m)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 0.4) {
		tmp = ((t_1 * 2.0) / k_m) * ((((l / k_m) / k_m) / k_m) / t);
	} else {
		tmp = (((2.0 * t_1) * (0.3333333333333333 * l)) / (k_m * 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) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * l
    if (k_m <= 0.4d0) then
        tmp = ((t_1 * 2.0d0) / k_m) * ((((l / k_m) / k_m) / k_m) / t)
    else
        tmp = (((2.0d0 * t_1) * (0.3333333333333333d0 * l)) / (k_m * 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 t_1 = Math.cos(k_m) * l;
	double tmp;
	if (k_m <= 0.4) {
		tmp = ((t_1 * 2.0) / k_m) * ((((l / k_m) / k_m) / k_m) / t);
	} else {
		tmp = (((2.0 * t_1) * (0.3333333333333333 * l)) / (k_m * k_m)) / t;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if k_m <= 0.4:
		tmp = ((t_1 * 2.0) / k_m) * ((((l / k_m) / k_m) / k_m) / t)
	else:
		tmp = (((2.0 * t_1) * (0.3333333333333333 * l)) / (k_m * k_m)) / t
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 0.4)
		tmp = Float64(Float64(Float64(t_1 * 2.0) / k_m) * Float64(Float64(Float64(Float64(l / k_m) / k_m) / k_m) / t));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * t_1) * Float64(0.3333333333333333 * l)) / Float64(k_m * k_m)) / t);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (k_m <= 0.4)
		tmp = ((t_1 * 2.0) / k_m) * ((((l / k_m) / k_m) / k_m) / t);
	else
		tmp = (((2.0 * t_1) * (0.3333333333333333 * l)) / (k_m * k_m)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.4], N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(0.3333333333333333 * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.40000000000000002

   1. Initial program 35.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.4

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

   3. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 74.8%

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

   9. Applied rewrites 84.0%

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

   11. Applied rewrites 95.0%

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

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 84.1%

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

   if 0.40000000000000002 < k

   1. Initial program 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 35.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 35.6

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

   3. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 63.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 20.7%

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

   13. Taylor expanded in k around inf

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

   15. Applied rewrites 48.4%

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

Alternative 9: 76.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\cos k\_m} \cdot \left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{elif}\;k\_m \leq 0.4:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{k\_m \cdot k\_m}}{k\_m \cdot k\_m} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \left(0.3333333333333333 \cdot \ell\right)}{k\_m \cdot k\_m}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e-105)
   (/ 2.0 (* (/ (* (* k_m k_m) t) (cos k_m)) (* (/ k_m l) (/ k_m l))))
   (if (<= k_m 0.4)
     (* (/ (/ (* l 2.0) (* k_m k_m)) (* k_m k_m)) (/ l t))
     (/
      (/ (* (* 2.0 (* (cos k_m) l)) (* 0.3333333333333333 l)) (* k_m k_m))
      t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e-105) {
		tmp = 2.0 / ((((k_m * k_m) * t) / cos(k_m)) * ((k_m / l) * (k_m / l)));
	} else if (k_m <= 0.4) {
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	} else {
		tmp = (((2.0 * (cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * 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 <= 2.2d-105) then
        tmp = 2.0d0 / ((((k_m * k_m) * t) / cos(k_m)) * ((k_m / l) * (k_m / l)))
    else if (k_m <= 0.4d0) then
        tmp = (((l * 2.0d0) / (k_m * k_m)) / (k_m * k_m)) * (l / t)
    else
        tmp = (((2.0d0 * (cos(k_m) * l)) * (0.3333333333333333d0 * l)) / (k_m * 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 <= 2.2e-105) {
		tmp = 2.0 / ((((k_m * k_m) * t) / Math.cos(k_m)) * ((k_m / l) * (k_m / l)));
	} else if (k_m <= 0.4) {
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	} else {
		tmp = (((2.0 * (Math.cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * k_m)) / t;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.2e-105:
		tmp = 2.0 / ((((k_m * k_m) * t) / math.cos(k_m)) * ((k_m / l) * (k_m / l)))
	elif k_m <= 0.4:
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t)
	else:
		tmp = (((2.0 * (math.cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * k_m)) / t
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e-105)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) / cos(k_m)) * Float64(Float64(k_m / l) * Float64(k_m / l))));
	elseif (k_m <= 0.4)
		tmp = Float64(Float64(Float64(Float64(l * 2.0) / Float64(k_m * k_m)) / Float64(k_m * k_m)) * Float64(l / t));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(cos(k_m) * l)) * Float64(0.3333333333333333 * l)) / Float64(k_m * 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 <= 2.2e-105)
		tmp = 2.0 / ((((k_m * k_m) * t) / cos(k_m)) * ((k_m / l) * (k_m / l)));
	elseif (k_m <= 0.4)
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	else
		tmp = (((2.0 * (cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * 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, 2.2e-105], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.4], N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.20000000000000004e-105

   1. Initial program 36.5%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 77.9%

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

   if 2.20000000000000004e-105 < k < 0.40000000000000002

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 30.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 30.1

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

   3. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 91.0%

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

   12. Applied rewrites 99.2%

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

   if 0.40000000000000002 < k

   1. Initial program 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 35.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 35.6

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

   3. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 63.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 20.7%

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

   13. Taylor expanded in k around inf

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

   15. Applied rewrites 48.4%

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

Alternative 10: 76.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 0.4:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{k\_m \cdot k\_m}}{k\_m \cdot k\_m} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \left(\cos k\_m \cdot \ell\right)\right) \cdot \left(0.3333333333333333 \cdot \ell\right)}{k\_m \cdot k\_m}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e-105)
   (* (/ 2.0 (* (* k_m k_m) t)) (* (/ l k_m) (/ l k_m)))
   (if (<= k_m 0.4)
     (* (/ (/ (* l 2.0) (* k_m k_m)) (* k_m k_m)) (/ l t))
     (/
      (/ (* (* 2.0 (* (cos k_m) l)) (* 0.3333333333333333 l)) (* k_m k_m))
      t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e-105) {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 0.4) {
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	} else {
		tmp = (((2.0 * (cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * 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 <= 2.2d-105) then
        tmp = (2.0d0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
    else if (k_m <= 0.4d0) then
        tmp = (((l * 2.0d0) / (k_m * k_m)) / (k_m * k_m)) * (l / t)
    else
        tmp = (((2.0d0 * (cos(k_m) * l)) * (0.3333333333333333d0 * l)) / (k_m * 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 <= 2.2e-105) {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	} else if (k_m <= 0.4) {
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	} else {
		tmp = (((2.0 * (Math.cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * k_m)) / t;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.2e-105:
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
	elif k_m <= 0.4:
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t)
	else:
		tmp = (((2.0 * (math.cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * k_m)) / t
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e-105)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	elseif (k_m <= 0.4)
		tmp = Float64(Float64(Float64(Float64(l * 2.0) / Float64(k_m * k_m)) / Float64(k_m * k_m)) * Float64(l / t));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(cos(k_m) * l)) * Float64(0.3333333333333333 * l)) / Float64(k_m * 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 <= 2.2e-105)
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	elseif (k_m <= 0.4)
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	else
		tmp = (((2.0 * (cos(k_m) * l)) * (0.3333333333333333 * l)) / (k_m * 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, 2.2e-105], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.4], N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.20000000000000004e-105

   1. Initial program 36.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 40.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 40.6

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

   3. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 77.4%

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

   if 2.20000000000000004e-105 < k < 0.40000000000000002

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 30.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 30.1

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

   3. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 91.0%

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

   12. Applied rewrites 99.2%

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

   if 0.40000000000000002 < k

   1. Initial program 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 35.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 35.6

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

   3. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 63.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 20.7%

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

   13. Taylor expanded in k around inf

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

   15. Applied rewrites 48.4%

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

Alternative 11: 72.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{k\_m \cdot k\_m}}{k\_m \cdot k\_m} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.2e+23)
   (* (/ (/ (* l 2.0) (* k_m k_m)) (* k_m k_m)) (/ l t))
   (* (/ 2.0 (* (* k_m k_m) t)) (* (/ l k_m) (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.2e+23) {
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / 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 (t <= 4.2d+23) then
        tmp = (((l * 2.0d0) / (k_m * k_m)) / (k_m * k_m)) * (l / t)
    else
        tmp = (2.0d0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / 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 (t <= 4.2e+23) {
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4.2e+23:
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t)
	else:
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.2e+23)
		tmp = Float64(Float64(Float64(Float64(l * 2.0) / Float64(k_m * k_m)) / Float64(k_m * k_m)) * Float64(l / t));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 4.2e+23)
		tmp = (((l * 2.0) / (k_m * k_m)) / (k_m * k_m)) * (l / t);
	else
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.2e+23], N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.2000000000000003e23

   1. Initial program 36.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 42.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 42.2

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

   3. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 67.7%

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

   12. Applied rewrites 72.1%

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

   if 4.2000000000000003e23 < t

   1. Initial program 24.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 26.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 26.7

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

   3. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 73.8%

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

Alternative 12: 69.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{2}{k\_m \cdot k\_m}\\ \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-164}:\\ \;\;\;\;\left(\frac{t\_1}{k\_m \cdot k\_m} \cdot \ell\right) \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* k_m k_m))))
   (if (<= l 1.62e-164)
     (* (* (/ t_1 (* k_m k_m)) l) (/ l t))
     (/ (* t_1 (* l l)) (* (* k_m k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 2.0 / (k_m * k_m);
	double tmp;
	if (l <= 1.62e-164) {
		tmp = ((t_1 / (k_m * k_m)) * l) * (l / t);
	} else {
		tmp = (t_1 * (l * l)) / ((k_m * 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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (k_m * k_m)
    if (l <= 1.62d-164) then
        tmp = ((t_1 / (k_m * k_m)) * l) * (l / t)
    else
        tmp = (t_1 * (l * l)) / ((k_m * 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 t_1 = 2.0 / (k_m * k_m);
	double tmp;
	if (l <= 1.62e-164) {
		tmp = ((t_1 / (k_m * k_m)) * l) * (l / t);
	} else {
		tmp = (t_1 * (l * l)) / ((k_m * k_m) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 2.0 / (k_m * k_m)
	tmp = 0
	if l <= 1.62e-164:
		tmp = ((t_1 / (k_m * k_m)) * l) * (l / t)
	else:
		tmp = (t_1 * (l * l)) / ((k_m * k_m) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(2.0 / Float64(k_m * k_m))
	tmp = 0.0
	if (l <= 1.62e-164)
		tmp = Float64(Float64(Float64(t_1 / Float64(k_m * k_m)) * l) * Float64(l / t));
	else
		tmp = Float64(Float64(t_1 * Float64(l * l)) / Float64(Float64(k_m * k_m) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 2.0 / (k_m * k_m);
	tmp = 0.0;
	if (l <= 1.62e-164)
		tmp = ((t_1 / (k_m * k_m)) * l) * (l / t);
	else
		tmp = (t_1 * (l * l)) / ((k_m * k_m) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.62e-164], N[(N[(N[(t$95$1 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.62000000000000005e-164

   1. Initial program 33.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.3

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

   3. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 69.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 68.1%

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

   12. Applied rewrites 68.1%

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

   if 1.62000000000000005e-164 < l

   1. Initial program 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 37.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 37.2

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

   3. Applied rewrites 39.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 55.6%

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

   9. Applied rewrites 55.6%

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

   11. Applied rewrites 67.3%

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

Alternative 13: 66.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-164}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot \left(\frac{\ell}{t} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m \cdot k\_m} \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1.62e-164)
   (* (/ 2.0 (* (* k_m k_m) (* k_m k_m))) (* (/ l t) l))
   (/ (* (/ 2.0 (* k_m k_m)) (* l l)) (* (* k_m k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.62e-164) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l);
	} else {
		tmp = ((2.0 / (k_m * k_m)) * (l * l)) / ((k_m * 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 (l <= 1.62d-164) then
        tmp = (2.0d0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l)
    else
        tmp = ((2.0d0 / (k_m * k_m)) * (l * l)) / ((k_m * 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 (l <= 1.62e-164) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l);
	} else {
		tmp = ((2.0 / (k_m * k_m)) * (l * l)) / ((k_m * k_m) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1.62e-164:
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l)
	else:
		tmp = ((2.0 / (k_m * k_m)) * (l * l)) / ((k_m * k_m) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 1.62e-164)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(k_m * k_m))) * Float64(Float64(l / t) * l));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k_m * k_m)) * Float64(l * l)) / Float64(Float64(k_m * k_m) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 1.62e-164)
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l);
	else
		tmp = ((2.0 / (k_m * k_m)) * (l * l)) / ((k_m * k_m) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1.62e-164], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.62000000000000005e-164

   1. Initial program 33.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 39.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.3

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

   3. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 59.1%

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

   9. Applied rewrites 59.1%

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

   11. Applied rewrites 65.5%

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

   if 1.62000000000000005e-164 < l

   1. Initial program 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 37.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 37.2

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

   3. Applied rewrites 39.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 55.6%

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

   9. Applied rewrites 55.6%

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

   11. Applied rewrites 67.3%

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

Alternative 14: 67.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot \left(\frac{\ell}{t} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k\_m} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.9e+65)
   (* (/ 2.0 (* (* k_m k_m) (* k_m k_m))) (* (/ l t) l))
   (* (/ -0.3333333333333333 k_m) (* (/ l t) (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.9e+65) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l);
	} else {
		tmp = (-0.3333333333333333 / k_m) * ((l / t) * (l / 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 <= 7.9d+65) then
        tmp = (2.0d0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l)
    else
        tmp = ((-0.3333333333333333d0) / k_m) * ((l / t) * (l / 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 <= 7.9e+65) {
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l);
	} else {
		tmp = (-0.3333333333333333 / k_m) * ((l / t) * (l / k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.9e+65:
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l)
	else:
		tmp = (-0.3333333333333333 / k_m) * ((l / t) * (l / k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.9e+65)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(k_m * k_m))) * Float64(Float64(l / t) * l));
	else
		tmp = Float64(Float64(-0.3333333333333333 / k_m) * Float64(Float64(l / t) * Float64(l / 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 <= 7.9e+65)
		tmp = (2.0 / ((k_m * k_m) * (k_m * k_m))) * ((l / t) * l);
	else
		tmp = (-0.3333333333333333 / k_m) * ((l / t) * (l / 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, 7.9e+65], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / k$95$m), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.8999999999999998e65

   1. Initial program 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 38.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 38.9

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

   3. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 62.3%

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

   9. Applied rewrites 62.3%

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

   11. Applied rewrites 67.6%

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

   if 7.8999999999999998e65 < k

   1. Initial program 32.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. +-rgt-identity N/A

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

      12. lower-/.f64 37.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 37.1

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

   3. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 6.4%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 45.2%

       \[\leadsto \frac{-0.3333333333333333}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]
   11. Step-by-step derivation

   12. Applied rewrites 48.5%

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

Alternative 15: 72.5% accurate, 8.6× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / 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 = (2.0d0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. +-rgt-identity N/A

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

   12. lower-/.f64 38.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 38.5

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

3. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0

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

7. Applied rewrites 70.9%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 69.0%

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

Alternative 16: 31.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.3333333333333333}{k\_m} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k\_m}\right) \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (-0.3333333333333333 / k_m) * ((l / t) * (l / 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 = ((-0.3333333333333333d0) / k_m) * ((l / t) * (l / k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (-0.3333333333333333 / k_m) * ((l / t) * (l / k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (-0.3333333333333333 / k_m) * ((l / t) * (l / k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(-0.3333333333333333 / k_m) * Float64(Float64(l / t) * Float64(l / k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (-0.3333333333333333 / k_m) * ((l / t) * (l / k_m));
end

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. +-rgt-identity N/A

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

   12. lower-/.f64 38.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 38.5

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

3. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

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

7. Applied rewrites 46.5%

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

8. Taylor expanded in k around inf

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

10. Applied rewrites 23.3%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]
11. Step-by-step derivation

12. Applied rewrites 24.2%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k}}\right) \]
13. Add Preprocessing

Alternative 17: 31.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.3333333333333333}{k\_m} \cdot \left(\ell \cdot \frac{\ell}{t \cdot k\_m}\right) \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (-0.3333333333333333 / k_m) * (l * (l / (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 = ((-0.3333333333333333d0) / k_m) * (l * (l / (t * k_m)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (-0.3333333333333333 / k_m) * (l * (l / (t * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (-0.3333333333333333 / k_m) * (l * (l / (t * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(-0.3333333333333333 / k_m) * Float64(l * Float64(l / Float64(t * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (-0.3333333333333333 / k_m) * (l * (l / (t * k_m)));
end

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. +-rgt-identity N/A

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

   12. lower-/.f64 38.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 38.5

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

3. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

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

7. Applied rewrites 46.5%

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

8. Taylor expanded in k around inf

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

10. Applied rewrites 23.3%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]
11. Step-by-step derivation

12. Applied rewrites 24.1%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t \cdot k}}\right) \]
13. Add Preprocessing

Alternative 18: 30.3% accurate, 12.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{t}}{k\_m \cdot k\_m} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.3333333333333333 * (((l * l) / t) / (k_m * 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 = (-0.3333333333333333d0) * (((l * l) / t) / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.3333333333333333 * (((l * l) / t) / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.3333333333333333 * (((l * l) / t) / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.3333333333333333 * Float64(Float64(Float64(l * l) / t) / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.3333333333333333 * (((l * l) / t) / (k_m * k_m));
end

Wolfram

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

Derivation

1. Initial program 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. +-rgt-identity N/A

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

   12. lower-/.f64 38.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 38.5

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

3. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

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

7. Applied rewrites 46.5%

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

8. Taylor expanded in k around inf

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

10. Applied rewrites 23.3%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]
11. Step-by-step derivation

12. Applied rewrites 23.0%

    \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{k \cdot k}} \]
13. Add Preprocessing

Alternative 19: 30.8% accurate, 13.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{\left(t \cdot \left(-k\_m\right)\right) \cdot k\_m} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * 0.3333333333333333) / ((t * -k_m) * 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) / ((t * -k_m) * 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) / ((t * -k_m) * k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) * 0.3333333333333333) / ((t * -k_m) * k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) * 0.3333333333333333) / Float64(Float64(t * Float64(-k_m)) * k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) * 0.3333333333333333) / ((t * -k_m) * 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[(t * (-k$95$m)), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. +-rgt-identity N/A

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

   12. lower-/.f64 38.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 38.5

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

3. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

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

7. Applied rewrites 46.5%

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

8. Taylor expanded in k around inf

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

10. Applied rewrites 23.3%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]
11. Step-by-step derivation

12. Applied rewrites 22.9%

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

13. Final simplification 22.9%

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

Alternative 20: 30.3% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k\_m \cdot k\_m\right) \cdot t} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / ((k_m * k_m) * t);
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) * (-0.3333333333333333d0)) / ((k_m * k_m) * t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / ((k_m * k_m) * t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) * -0.3333333333333333) / ((k_m * k_m) * t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(Float64(k_m * k_m) * t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) * -0.3333333333333333) / ((k_m * k_m) * t);
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 * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. +-rgt-identity N/A

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

   12. lower-/.f64 38.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 38.5

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

3. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

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

7. Applied rewrites 46.5%

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

8. Taylor expanded in k around inf

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

10. Applied rewrites 23.3%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]
11. Step-by-step derivation

12. Applied rewrites 22.8%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(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))))