Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 99.4%

Time: 13.7s

Alternatives: 12

Speedup: 9.6×

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

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: 35.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

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: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 2.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k\_m \cdot \left(\tan k\_m \cdot k\_m\right)}{\ell} \cdot t\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 2.15e-21)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (/ (* (sin k_m) (* (tan k_m) k_m)) l) t) (/ 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 <= 2.15e-21) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / ((((sin(k_m) * (tan(k_m) * k_m)) / l) * t) * (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 <= 2.15d-21) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / ((((sin(k_m) * (tan(k_m) * k_m)) / l) * t) * (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 <= 2.15e-21) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / ((((Math.sin(k_m) * (Math.tan(k_m) * k_m)) / l) * t) * (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 <= 2.15e-21:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / ((((math.sin(k_m) * (math.tan(k_m) * k_m)) / l) * t) * (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 <= 2.15e-21)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k_m) * Float64(tan(k_m) * k_m)) / l) * t) * Float64(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 <= 2.15e-21)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / ((((sin(k_m) * (tan(k_m) * k_m)) / l) * t) * (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, 2.15e-21], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1499999999999999e-21

   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 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 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 83.4%

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

   if 2.1499999999999999e-21 < k

   1. Initial program 33.2%

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

   3. Taylor expanded in t around 0

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

      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 98.1%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.5%

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

   11. Applied rewrites 99.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \left(\tan k \cdot k\right)}{\ell} \cdot t\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k\_m \cdot t}{\ell} \cdot \tan k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= (* l l) 0.0)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (* (/ (* (sin k_m) t) l) (tan k_m)) k_m) (/ 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 ((l * l) <= 0.0) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((((sin(k_m) * t) / l) * tan(k_m)) * k_m) * (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 ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / (((((sin(k_m) * t) / l) * tan(k_m)) * k_m) * (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 ((l * l) <= 0.0) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((((Math.sin(k_m) * t) / l) * Math.tan(k_m)) * k_m) * (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 (l * l) <= 0.0:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / (((((math.sin(k_m) * t) / l) * math.tan(k_m)) * k_m) * (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 (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k_m) * t) / l) * tan(k_m)) * k_m) * Float64(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 ((l * l) <= 0.0)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / (((((sin(k_m) * t) / l) * tan(k_m)) * k_m) * (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[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 98.7%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 44.2%

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

   3. Taylor expanded in t around 0

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

      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 96.3%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 98.9%

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

   11. Applied rewrites 97.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-289}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(t \cdot k\_m\right) \cdot \sin k\_m}{\ell} \cdot \tan k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= (* l l) 1e-289)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (/ (* (* t k_m) (sin k_m)) l) (tan k_m)) (/ 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 ((l * l) <= 1e-289) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((((t * k_m) * sin(k_m)) / l) * tan(k_m)) * (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 ((l * l) <= 1d-289) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / (((((t * k_m) * sin(k_m)) / l) * tan(k_m)) * (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 ((l * l) <= 1e-289) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((((t * k_m) * Math.sin(k_m)) / l) * Math.tan(k_m)) * (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 (l * l) <= 1e-289:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / (((((t * k_m) * math.sin(k_m)) / l) * math.tan(k_m)) * (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 (Float64(l * l) <= 1e-289)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * k_m) * sin(k_m)) / l) * tan(k_m)) * Float64(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 ((l * l) <= 1e-289)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / (((((t * k_m) * sin(k_m)) / l) * tan(k_m)) * (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[N[(l * l), $MachinePrecision], 1e-289], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * k$95$m), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-289

   1. Initial program 26.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 68.8

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 98.8%

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

   if 1e-289 < (*.f64 l l)

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 96.1%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 96.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-289}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(t \cdot k\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-212}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right) \cdot {\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= (* l l) 4e-212)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (/ k_m l) (* t (/ k_m l))) (pow (sin k_m) 2.0))))))

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 ((l * l) <= 4e-212) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((k_m / l) * (t * (k_m / l))) * pow(sin(k_m), 2.0));
	}
	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 ((l * l) <= 4d-212) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / (((k_m / l) * (t * (k_m / l))) * (sin(k_m) ** 2.0d0))
    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 ((l * l) <= 4e-212) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((k_m / l) * (t * (k_m / l))) * Math.pow(Math.sin(k_m), 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if (l * l) <= 4e-212:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / (((k_m / l) * (t * (k_m / l))) * math.pow(math.sin(k_m), 2.0))
	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 (Float64(l * l) <= 4e-212)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(t * Float64(k_m / l))) * (sin(k_m) ^ 2.0)));
	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 ((l * l) <= 4e-212)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / (((k_m / l) * (t * (k_m / l))) * (sin(k_m) ^ 2.0));
	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[N[(l * l), $MachinePrecision], 4e-212], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 3.99999999999999982e-212

   1. Initial program 25.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 96.7%

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

   if 3.99999999999999982e-212 < (*.f64 l l)

   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 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 95.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 70.8%

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

   10. Applied rewrites 71.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-212}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-212}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k\_m}^{2} \cdot t}{\ell} \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= (* l l) 4e-212)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (/ (* (pow (sin k_m) 2.0) t) l) k_m) (/ 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 ((l * l) <= 4e-212) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / ((((pow(sin(k_m), 2.0) * t) / l) * k_m) * (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 ((l * l) <= 4d-212) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / (((((sin(k_m) ** 2.0d0) * t) / l) * k_m) * (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 ((l * l) <= 4e-212) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / ((((Math.pow(Math.sin(k_m), 2.0) * t) / l) * k_m) * (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 (l * l) <= 4e-212:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / ((((math.pow(math.sin(k_m), 2.0) * t) / l) * k_m) * (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 (Float64(l * l) <= 4e-212)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) / l) * k_m) * Float64(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 ((l * l) <= 4e-212)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / (((((sin(k_m) ^ 2.0) * t) / l) * k_m) * (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[N[(l * l), $MachinePrecision], 4e-212], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 3.99999999999999982e-212

   1. Initial program 25.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 96.7%

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

   if 3.99999999999999982e-212 < (*.f64 l l)

   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 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 95.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 70.8%

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

   10. Applied rewrites 71.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-212}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\ell} \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 2.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot t\right) \cdot \left(\frac{\tan k\_m}{\ell} \cdot k\_m\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 2.15e-21)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (* (sin k_m) t) (* (/ (tan k_m) l) k_m)) (/ 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 <= 2.15e-21) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((sin(k_m) * t) * ((tan(k_m) / l) * k_m)) * (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 <= 2.15d-21) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / (((sin(k_m) * t) * ((tan(k_m) / l) * k_m)) * (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 <= 2.15e-21) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((Math.sin(k_m) * t) * ((Math.tan(k_m) / l) * k_m)) * (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 <= 2.15e-21:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / (((math.sin(k_m) * t) * ((math.tan(k_m) / l) * k_m)) * (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 <= 2.15e-21)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * t) * Float64(Float64(tan(k_m) / l) * k_m)) * Float64(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 <= 2.15e-21)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / (((sin(k_m) * t) * ((tan(k_m) / l) * k_m)) * (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, 2.15e-21], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1499999999999999e-21

   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 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 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 83.4%

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

   if 2.1499999999999999e-21 < k

   1. Initial program 33.2%

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

   3. Taylor expanded in t around 0

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

      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 98.1%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.5%

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

   11. Applied rewrites 99.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{\tan k}{\ell} \cdot k\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 2.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \frac{k\_m}{\ell}\right) \cdot t\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 2.15e-21)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (* (* (sin k_m) (tan k_m)) (/ k_m l)) t) (/ 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 <= 2.15e-21) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / ((((sin(k_m) * tan(k_m)) * (k_m / l)) * t) * (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 <= 2.15d-21) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / ((((sin(k_m) * tan(k_m)) * (k_m / l)) * t) * (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 <= 2.15e-21) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / ((((Math.sin(k_m) * Math.tan(k_m)) * (k_m / l)) * t) * (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 <= 2.15e-21:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / ((((math.sin(k_m) * math.tan(k_m)) * (k_m / l)) * t) * (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 <= 2.15e-21)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(k_m / l)) * t) * Float64(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 <= 2.15e-21)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / ((((sin(k_m) * tan(k_m)) * (k_m / l)) * t) * (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, 2.15e-21], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1499999999999999e-21

   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 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 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 83.4%

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

   if 2.1499999999999999e-21 < k

   1. Initial program 33.2%

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

   3. Taylor expanded in t around 0

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

      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 98.1%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 1.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 5.0)
     (/ 2.0 (* (* t t_1) t_1))
     (/ 2.0 (* (* (* (pow (sin k_m) 2.0) (/ k_m l)) t) (/ 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 <= 5.0) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((pow(sin(k_m), 2.0) * (k_m / l)) * t) * (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 <= 5.0d0) then
        tmp = 2.0d0 / ((t * t_1) * t_1)
    else
        tmp = 2.0d0 / ((((sin(k_m) ** 2.0d0) * (k_m / l)) * t) * (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 <= 5.0) {
		tmp = 2.0 / ((t * t_1) * t_1);
	} else {
		tmp = 2.0 / (((Math.pow(Math.sin(k_m), 2.0) * (k_m / l)) * t) * (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 <= 5.0:
		tmp = 2.0 / ((t * t_1) * t_1)
	else:
		tmp = 2.0 / (((math.pow(math.sin(k_m), 2.0) * (k_m / l)) * t) * (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 <= 5.0)
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k_m) ^ 2.0) * Float64(k_m / l)) * t) * Float64(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 <= 5.0)
		tmp = 2.0 / ((t * t_1) * t_1);
	else
		tmp = 2.0 / ((((sin(k_m) ^ 2.0) * (k_m / l)) * t) * (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, 5.0], N[(2.0 / N[(N[(t * t$95$1), $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[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5

   1. Initial program 39.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 70.9

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 75.7%

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

   9. Applied rewrites 83.5%

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

   if 5 < k

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 97.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 56.2%

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

   10. Applied rewrites 56.2%

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

4. Final simplification 77.7%

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

Alternative 9: 70.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{t \cdot k\_m}\\ t_2 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+255}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_2\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{0.5}{t\_1} - \frac{\cos \left(k\_m + k\_m\right) \cdot 0.5}{t\_1}\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* t k_m))) (t_2 (* (/ k_m l) k_m)))
   (if (<= (* l l) 2e+255)
     (/ 2.0 (* (* t t_2) t_2))
     (/ 2.0 (* (- (/ 0.5 t_1) (/ (* (cos (+ k_m k_m)) 0.5) t_1)) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (t * k_m);
	double t_2 = (k_m / l) * k_m;
	double tmp;
	if ((l * l) <= 2e+255) {
		tmp = 2.0 / ((t * t_2) * t_2);
	} else {
		tmp = 2.0 / (((0.5 / t_1) - ((cos((k_m + k_m)) * 0.5) / t_1)) * (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) :: t_2
    real(8) :: tmp
    t_1 = l / (t * k_m)
    t_2 = (k_m / l) * k_m
    if ((l * l) <= 2d+255) then
        tmp = 2.0d0 / ((t * t_2) * t_2)
    else
        tmp = 2.0d0 / (((0.5d0 / t_1) - ((cos((k_m + k_m)) * 0.5d0) / t_1)) * (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 = l / (t * k_m);
	double t_2 = (k_m / l) * k_m;
	double tmp;
	if ((l * l) <= 2e+255) {
		tmp = 2.0 / ((t * t_2) * t_2);
	} else {
		tmp = 2.0 / (((0.5 / t_1) - ((Math.cos((k_m + k_m)) * 0.5) / t_1)) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (t * k_m)
	t_2 = (k_m / l) * k_m
	tmp = 0
	if (l * l) <= 2e+255:
		tmp = 2.0 / ((t * t_2) * t_2)
	else:
		tmp = 2.0 / (((0.5 / t_1) - ((math.cos((k_m + k_m)) * 0.5) / t_1)) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(t * k_m))
	t_2 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e+255)
		tmp = Float64(2.0 / Float64(Float64(t * t_2) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 / t_1) - Float64(Float64(cos(Float64(k_m + k_m)) * 0.5) / t_1)) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (t * k_m);
	t_2 = (k_m / l) * k_m;
	tmp = 0.0;
	if ((l * l) <= 2e+255)
		tmp = 2.0 / ((t * t_2) * t_2);
	else
		tmp = 2.0 / (((0.5 / t_1) - ((cos((k_m + k_m)) * 0.5) / t_1)) * (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[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 2e+255], N[(2.0 / N[(N[(t * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 / t$95$1), $MachinePrecision] - N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999998e255

   1. Initial program 38.4%

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

   3. Taylor expanded in 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 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 74.2%

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

   9. Applied rewrites 83.6%

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

   if 1.99999999999999998e255 < (*.f64 l l)

   1. Initial program 37.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 92.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 67.5%

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

   10. Applied rewrites 28.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+255}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{0.5}{\frac{\ell}{t \cdot k}} - \frac{\cos \left(k + k\right) \cdot 0.5}{\frac{\ell}{t \cdot k}}\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \frac{2}{\left(t \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	return 2.0 / ((t * t_1) * t_1);
}

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
    t_1 = (k_m / l) * k_m
    code = 2.0d0 / ((t * t_1) * t_1)
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;
	return 2.0 / ((t * t_1) * t_1);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	return 2.0 / ((t * t_1) * t_1)

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	return Float64(2.0 / Float64(Float64(t * t_1) * t_1))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 2.0 / ((t * t_1) * t_1);
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]}, N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in 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 66.6

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

5. Applied rewrites 66.6%

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

7. Applied rewrites 70.3%

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

9. Applied rewrites 76.5%

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

10. Final simplification 76.5%

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

Alternative 11: 75.3% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

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 = 2.0d0 / ((((t * k_m) * (k_m / l)) * (k_m / l)) * k_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in 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 66.6

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

5. Applied rewrites 66.6%

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

7. Applied rewrites 70.3%

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

9. Applied rewrites 72.2%

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

11. Applied rewrites 73.9%

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

12. Final simplification 73.9%

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

Alternative 12: 65.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

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 = 2.0d0 / ((((k_m * k_m) * (t * k_m)) / (l * l)) * k_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in 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 66.6

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

5. Applied rewrites 66.6%

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

7. Applied rewrites 70.3%

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

9. Applied rewrites 72.2%

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

11. Applied rewrites 64.8%

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

12. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2024278 
(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))))