Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.3% → 99.2%

Time: 13.2s

Alternatives: 12

Speedup: 8.6×

• 39.5% 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

real(8) function code(t, l, k)
    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: 36.3% 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

real(8) function code(t, l, k)
    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: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k\_m}^{2} \cdot k\_m}{\ell} \cdot t\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 2.2e-83)
     (/ 2.0 (* (* t_1 t) t_1))
     (/
      2.0
      (* (* (/ (* (pow (sin k_m) 2.0) k_m) l) t) (/ (/ k_m (cos k_m)) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 2.2e-83) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((pow(sin(k_m), 2.0) * k_m) / l) * t) * ((k_m / cos(k_m)) / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 2.2d-83) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((((sin(k_m) ** 2.0d0) * k_m) / l) * t) * ((k_m / cos(k_m)) / l))
    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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 2.2e-83) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((Math.pow(Math.sin(k_m), 2.0) * k_m) / l) * t) * ((k_m / Math.cos(k_m)) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 2.2e-83:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((math.pow(math.sin(k_m), 2.0) * k_m) / l) * t) * ((k_m / math.cos(k_m)) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 2.2e-83)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((sin(k_m) ^ 2.0) * k_m) / l) * t) * Float64(Float64(k_m / cos(k_m)) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 2.2e-83)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((((sin(k_m) ^ 2.0) * k_m) / l) * t) * ((k_m / cos(k_m)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 2.2e-83], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000008e-83

   1. Initial program 44.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 69.6%

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

   11. Applied rewrites 84.8%

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

   if 2.20000000000000008e-83 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k}^{2} \cdot k}{\ell} \cdot t\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k\_m}^{2} \cdot \frac{k\_m}{\ell}\right) \cdot t\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 2.2e-83)
     (/ 2.0 (* (* t_1 t) t_1))
     (/
      2.0
      (* (* (* (pow (sin k_m) 2.0) (/ k_m l)) t) (/ (/ k_m (cos k_m)) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 2.2e-83) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((pow(sin(k_m), 2.0) * (k_m / l)) * t) * ((k_m / cos(k_m)) / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 2.2d-83) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((sin(k_m) ** 2.0d0) * (k_m / l)) * t) * ((k_m / cos(k_m)) / l))
    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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 2.2e-83) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((Math.pow(Math.sin(k_m), 2.0) * (k_m / l)) * t) * ((k_m / Math.cos(k_m)) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 2.2e-83:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((math.pow(math.sin(k_m), 2.0) * (k_m / l)) * t) * ((k_m / math.cos(k_m)) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 2.2e-83)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k_m) ^ 2.0) * Float64(k_m / l)) * t) * Float64(Float64(k_m / cos(k_m)) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 2.2e-83)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((sin(k_m) ^ 2.0) * (k_m / l)) * t) * ((k_m / cos(k_m)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 2.2e-83], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000008e-83

   1. Initial program 44.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 69.6%

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

   11. Applied rewrites 84.8%

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

   if 2.20000000000000008e-83 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 99.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \left(\left({\sin k\_m}^{2} \cdot \frac{k\_m}{\ell}\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 2.2e-83)
     (/ 2.0 (* (* t_1 t) t_1))
     (/
      2.0
      (* (/ k_m (* (cos k_m) l)) (* (* (pow (sin k_m) 2.0) (/ k_m l)) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 2.2e-83) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((k_m / (cos(k_m) * l)) * ((pow(sin(k_m), 2.0) * (k_m / l)) * t));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 2.2d-83) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((k_m / (cos(k_m) * l)) * (((sin(k_m) ** 2.0d0) * (k_m / l)) * 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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 2.2e-83) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((k_m / (Math.cos(k_m) * l)) * ((Math.pow(Math.sin(k_m), 2.0) * (k_m / l)) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 2.2e-83:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((k_m / (math.cos(k_m) * l)) * ((math.pow(math.sin(k_m), 2.0) * (k_m / l)) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 2.2e-83)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(Float64((sin(k_m) ^ 2.0) * Float64(k_m / l)) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 2.2e-83)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((k_m / (cos(k_m) * l)) * (((sin(k_m) ^ 2.0) * (k_m / l)) * 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[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 2.2e-83], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000008e-83

   1. Initial program 44.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 69.6%

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

   11. Applied rewrites 84.8%

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

   if 2.20000000000000008e-83 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k \cdot \ell} \cdot \left(\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \frac{k\_m}{\ell}\right) \cdot t\right) \cdot \frac{k\_m}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 9.6e-5)
     (/ 2.0 (* (* t_1 t) t_1))
     (/
      2.0
      (*
       (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (/ k_m l)) t)
       (/ k_m (* (cos k_m) l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 9.6e-5) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m / l)) * t) * (k_m / (cos(k_m) * l)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 9.6d-5) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * (k_m / l)) * t) * (k_m / (cos(k_m) * l)))
    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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 9.6e-5) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * (k_m / l)) * t) * (k_m / (Math.cos(k_m) * l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 9.6e-5:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * (k_m / l)) * t) * (k_m / (math.cos(k_m) * l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 9.6e-5)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m / l)) * t) * Float64(k_m / Float64(cos(k_m) * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 9.6e-5)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m / l)) * t) * (k_m / (cos(k_m) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 9.6e-5], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.6000000000000002e-5

   1. Initial program 43.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 74.4%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 86.1%

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

   if 9.6000000000000002e-5 < k

   1. Initial program 28.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

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

   11. Applied rewrites 99.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{k}{\cos k \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.000135:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot \frac{\cos k\_m \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot \left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.000135)
     (/ 2.0 (* (* t_1 t) t_1))
     (*
      (/ (* l l) t)
      (/
       (* (cos k_m) 2.0)
       (* (* k_m k_m) (- 0.5 (* (cos (+ k_m k_m)) 0.5))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000135) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = ((l * l) / t) * ((cos(k_m) * 2.0) / ((k_m * k_m) * (0.5 - (cos((k_m + k_m)) * 0.5))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 0.000135d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = ((l * l) / t) * ((cos(k_m) * 2.0d0) / ((k_m * k_m) * (0.5d0 - (cos((k_m + k_m)) * 0.5d0))))
    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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000135) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = ((l * l) / t) * ((Math.cos(k_m) * 2.0) / ((k_m * k_m) * (0.5 - (Math.cos((k_m + k_m)) * 0.5))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 0.000135:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = ((l * l) / t) * ((math.cos(k_m) * 2.0) / ((k_m * k_m) * (0.5 - (math.cos((k_m + k_m)) * 0.5))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.000135)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(Float64(Float64(l * l) / t) * Float64(Float64(cos(k_m) * 2.0) / Float64(Float64(k_m * k_m) * Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 0.000135)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = ((l * l) / t) * ((cos(k_m) * 2.0) / ((k_m * k_m) * (0.5 - (cos((k_m + k_m)) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.000135], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000002e-4

   1. Initial program 43.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 74.4%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 86.1%

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

   if 1.35000000000000002e-4 < k

   1. Initial program 28.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 64.9%

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

   10. Applied rewrites 64.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000135:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot \frac{\cos k \cdot 2}{\left(k \cdot k\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{elif}\;k\_m \leq 1.46 \cdot 10^{+44}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot k\_m, k\_m, 2\right) \cdot \ell\right)\right) \cdot {k\_m}^{-4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \frac{k\_m}{\ell}\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 6e-35)
     (/ 2.0 (* (* t_1 t) t_1))
     (if (<= k_m 1.46e+44)
       (*
        (* (/ l t) (* (fma (* -0.3333333333333333 k_m) k_m 2.0) l))
        (pow k_m -4.0))
       (/
        2.0
        (*
         (/ k_m l)
         (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (/ k_m l)) t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 6e-35) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else if (k_m <= 1.46e+44) {
		tmp = ((l / t) * (fma((-0.3333333333333333 * k_m), k_m, 2.0) * l)) * pow(k_m, -4.0);
	} else {
		tmp = 2.0 / ((k_m / l) * (((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m / l)) * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 6e-35)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	elseif (k_m <= 1.46e+44)
		tmp = Float64(Float64(Float64(l / t) * Float64(fma(Float64(-0.3333333333333333 * k_m), k_m, 2.0) * l)) * (k_m ^ -4.0));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m / l)) * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 6e-35], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.46e+44], N[(N[(N[(l / t), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * k$95$m), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.99999999999999978e-35

   1. Initial program 44.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 85.7%

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

   if 5.99999999999999978e-35 < k < 1.4599999999999999e44

   1. Initial program 14.6%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 73.4%

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

   10. Applied rewrites 75.1%

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

   if 1.4599999999999999e44 < k

   1. Initial program 30.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 63.7%

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

   12. Applied rewrites 63.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.46 \cdot 10^{+44}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot k, k, 2\right) \cdot \ell\right)\right) \cdot {k}^{-4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \frac{k}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k\_m}{\cos k\_m}}{\ell} \cdot \frac{k\_m}{\ell}\right) \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.000185)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (/ (/ k_m (cos k_m)) l) (/ k_m l)) (* (* k_m k_m) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k_m / cos(k_m)) / l) * (k_m / l)) * ((k_m * k_m) * t));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 0.000185d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((k_m / cos(k_m)) / l) * (k_m / 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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k_m / Math.cos(k_m)) / l) * (k_m / l)) * ((k_m * k_m) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 0.000185:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((k_m / math.cos(k_m)) / l) * (k_m / l)) * ((k_m * k_m) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.000185)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / cos(k_m)) / l) * Float64(k_m / 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 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 0.000185)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((k_m / cos(k_m)) / l) * (k_m / 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[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.000185], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85e-4

   1. Initial program 43.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 74.4%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 86.1%

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

   if 1.85e-4 < k

   1. Initial program 28.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.4%

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

   10. Applied rewrites 60.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot k\_m}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.000185)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (/ (* (* (* (* k_m k_m) t) (/ k_m l)) k_m) (* (cos k_m) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (cos(k_m) * l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 0.000185d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (cos(k_m) * l))
    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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (Math.cos(k_m) * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 0.000185:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (math.cos(k_m) * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.000185)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m / l)) * k_m) / Float64(cos(k_m) * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 0.000185)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (cos(k_m) * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.000185], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85e-4

   1. Initial program 43.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 74.4%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 86.1%

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

   if 1.85e-4 < k

   1. Initial program 28.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.4%

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

   10. Applied rewrites 60.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot k}{\cos k \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{-0.3333333333333333 \cdot \ell}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.000185)
     (/ 2.0 (* (* t_1 t) t_1))
     (* (/ l (* k_m k_m)) (/ (* -0.3333333333333333 l) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    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 = (k_m / l) * k_m
    if (k_m <= 0.000185d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = (l / (k_m * k_m)) * (((-0.3333333333333333d0) * l) / 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 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if k_m <= 0.000185:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.000185)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(-0.3333333333333333 * l) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (k_m <= 0.000185)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / 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[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.000185], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.3333333333333333 * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85e-4

   1. Initial program 43.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 74.4%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 86.1%

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

   if 1.85e-4 < k

   1. Initial program 28.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 58.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{-0.3333333333333333 \cdot \ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot \frac{k\_m}{\ell}\right) \cdot t\right) \cdot \frac{k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{-0.3333333333333333 \cdot \ell}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.000185)
   (/ 2.0 (* (* (* (* k_m k_m) (/ k_m l)) t) (/ k_m l)))
   (* (/ l (* k_m k_m)) (/ (* -0.3333333333333333 l) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((((k_m * k_m) * (k_m / l)) * t) * (k_m / l));
	} else {
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.000185d0) then
        tmp = 2.0d0 / ((((k_m * k_m) * (k_m / l)) * t) * (k_m / l))
    else
        tmp = (l / (k_m * k_m)) * (((-0.3333333333333333d0) * l) / 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 <= 0.000185) {
		tmp = 2.0 / ((((k_m * k_m) * (k_m / l)) * t) * (k_m / l));
	} else {
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.000185:
		tmp = 2.0 / ((((k_m * k_m) * (k_m / l)) * t) * (k_m / l))
	else:
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.000185)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * Float64(k_m / l)) * t) * Float64(k_m / l)));
	else
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(-0.3333333333333333 * l) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.000185)
		tmp = 2.0 / ((((k_m * k_m) * (k_m / l)) * t) * (k_m / l));
	else
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.000185], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.3333333333333333 * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85e-4

   1. Initial program 43.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 96.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 84.2%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 82.8%

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

   if 1.85e-4 < k

   1. Initial program 28.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 58.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{-0.3333333333333333 \cdot \ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m \cdot k\_m}{\ell \cdot \ell} \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{-0.3333333333333333 \cdot \ell}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.000185)
   (/ 2.0 (* (* (/ (* k_m k_m) (* l l)) (* k_m k_m)) t))
   (* (/ l (* k_m k_m)) (/ (* -0.3333333333333333 l) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.000185) {
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t);
	} else {
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.000185d0) then
        tmp = 2.0d0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t)
    else
        tmp = (l / (k_m * k_m)) * (((-0.3333333333333333d0) * l) / 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 <= 0.000185) {
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t);
	} else {
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.000185:
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t)
	else:
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.000185)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) / Float64(l * l)) * Float64(k_m * k_m)) * t));
	else
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(-0.3333333333333333 * l) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.000185)
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t);
	else
		tmp = (l / (k_m * k_m)) * ((-0.3333333333333333 * l) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.000185], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.3333333333333333 * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85e-4

   1. Initial program 43.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 74.4%

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

   if 1.85e-4 < k

   1. Initial program 28.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 58.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000185:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{-0.3333333333333333 \cdot \ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.3% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l / (k_m * k_m)) * (((-0.3333333333333333d0) * l) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 91.6%

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

6. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 49.7%

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

9. Taylor expanded in k around inf

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

11. Applied rewrites 31.0%

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

12. Final simplification 31.0%

    \[\leadsto \frac{\ell}{k \cdot k} \cdot \frac{-0.3333333333333333 \cdot \ell}{t} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(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))))