Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 99.3%

Time: 12.3s

Alternatives: 10

Speedup: 9.6×

• 39.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 35.6% 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: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t}{\frac{\frac{\ell}{k\_m}}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m}{\cos k\_m}}{\ell} \cdot \left(t \cdot \left({\sin k\_m}^{2} \cdot \frac{k\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e-58)
   (/ 2.0 (/ (* (* (/ k_m l) k_m) t) (/ (/ l k_m) k_m)))
   (/
    2.0
    (* (/ (/ k_m (cos k_m)) l) (* t (* (pow (sin k_m) 2.0) (/ k_m l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-58) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / (((k_m / cos(k_m)) / l) * (t * (pow(sin(k_m), 2.0) * (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) :: tmp
    if (k_m <= 5d-58) then
        tmp = 2.0d0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
    else
        tmp = 2.0d0 / (((k_m / cos(k_m)) / l) * (t * ((sin(k_m) ** 2.0d0) * (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 tmp;
	if (k_m <= 5e-58) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / (((k_m / Math.cos(k_m)) / l) * (t * (Math.pow(Math.sin(k_m), 2.0) * (k_m / l))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5e-58:
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
	else:
		tmp = 2.0 / (((k_m / math.cos(k_m)) / l) * (t * (math.pow(math.sin(k_m), 2.0) * (k_m / l))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-58)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / l) * k_m) * t) / Float64(Float64(l / k_m) / k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / cos(k_m)) / l) * Float64(t * Float64((sin(k_m) ^ 2.0) * Float64(k_m / l)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e-58)
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	else
		tmp = 2.0 / (((k_m / cos(k_m)) / l) * (t * ((sin(k_m) ^ 2.0) * (k_m / l))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e-58], N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999977e-58

   1. Initial program 41.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 86.4%

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

   if 4.99999999999999977e-58 < k

   1. Initial program 32.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 95.2%

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

   7. Applied rewrites 99.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\frac{\ell}{k\_m}} \cdot k\_m}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(t\_1 \cdot t\right) \cdot k\_m} \cdot \frac{\ell \cdot \ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\_1\right) \cdot \left(\frac{k\_m}{\ell} \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= (* l l) 2e-243)
     (/ 2.0 (* (/ (* t (* (/ k_m l) k_m)) (/ l k_m)) k_m))
     (if (<= (* l l) 4e+244)
       (* (/ (* 2.0 (cos k_m)) (* (* t_1 t) k_m)) (/ (* l l) k_m))
       (/ 2.0 (* (* t t_1) (* (/ k_m l) (/ (/ k_m (cos k_m)) l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double tmp;
	if ((l * l) <= 2e-243) {
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	} else if ((l * l) <= 4e+244) {
		tmp = ((2.0 * cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m);
	} else {
		tmp = 2.0 / ((t * t_1) * ((k_m / l) * ((k_m / cos(k_m)) / l)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    if ((l * l) <= 2d-243) then
        tmp = 2.0d0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m)
    else if ((l * l) <= 4d+244) then
        tmp = ((2.0d0 * cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m)
    else
        tmp = 2.0d0 / ((t * t_1) * ((k_m / l) * ((k_m / cos(k_m)) / l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if ((l * l) <= 2e-243) {
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	} else if ((l * l) <= 4e+244) {
		tmp = ((2.0 * Math.cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m);
	} else {
		tmp = 2.0 / ((t * t_1) * ((k_m / l) * ((k_m / Math.cos(k_m)) / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if (l * l) <= 2e-243:
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m)
	elif (l * l) <= 4e+244:
		tmp = ((2.0 * math.cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m)
	else:
		tmp = 2.0 / ((t * t_1) * ((k_m / l) * ((k_m / math.cos(k_m)) / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 2e-243)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(k_m / l) * k_m)) / Float64(l / k_m)) * k_m));
	elseif (Float64(l * l) <= 4e+244)
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(t_1 * t) * k_m)) * Float64(Float64(l * l) / k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * Float64(Float64(k_m / l) * Float64(Float64(k_m / cos(k_m)) / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if ((l * l) <= 2e-243)
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	elseif ((l * l) <= 4e+244)
		tmp = ((2.0 * cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m);
	else
		tmp = 2.0 / ((t * t_1) * ((k_m / l) * ((k_m / cos(k_m)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 2e-243], N[(2.0 / N[(N[(N[(t * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 4e+244], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.99999999999999999e-243

   1. Initial program 23.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 81.3%

      \[\leadsto \frac{2}{\frac{{\ell}^{-1} \cdot {\left({k}^{-4}\right)}^{-1}}{\ell} \cdot t} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.9%

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

   11. Applied rewrites 96.5%

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

   if 1.99999999999999999e-243 < (*.f64 l l) < 4.0000000000000003e244

   1. Initial program 44.4%

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

   3. Taylor expanded in 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 99.0%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 4.0000000000000003e244 < (*.f64 l l)

   1. Initial program 47.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 88.0%

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

   7. Applied rewrites 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot k\right)}{\frac{\ell}{k}} \cdot k}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left({\sin k}^{2} \cdot t\right) \cdot k} \cdot \frac{\ell \cdot \ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\cos k}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\frac{\ell}{k\_m}} \cdot k\_m}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+254}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(t\_1 \cdot t\right) \cdot k\_m} \cdot \frac{\ell \cdot \ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t\_1 \cdot k\_m}{\ell} \cdot \frac{k\_m}{\cos k\_m \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= (* l l) 2e-243)
     (/ 2.0 (* (/ (* t (* (/ k_m l) k_m)) (/ l k_m)) k_m))
     (if (<= (* l l) 1e+254)
       (* (/ (* 2.0 (cos k_m)) (* (* t_1 t) k_m)) (/ (* l l) k_m))
       (/ 2.0 (* t (* (/ (* t_1 k_m) l) (/ k_m (* (cos k_m) l)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double tmp;
	if ((l * l) <= 2e-243) {
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	} else if ((l * l) <= 1e+254) {
		tmp = ((2.0 * cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m);
	} else {
		tmp = 2.0 / (t * (((t_1 * k_m) / l) * (k_m / (cos(k_m) * l))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    if ((l * l) <= 2d-243) then
        tmp = 2.0d0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m)
    else if ((l * l) <= 1d+254) then
        tmp = ((2.0d0 * cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m)
    else
        tmp = 2.0d0 / (t * (((t_1 * k_m) / l) * (k_m / (cos(k_m) * l))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if ((l * l) <= 2e-243) {
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	} else if ((l * l) <= 1e+254) {
		tmp = ((2.0 * Math.cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m);
	} else {
		tmp = 2.0 / (t * (((t_1 * k_m) / l) * (k_m / (Math.cos(k_m) * l))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if (l * l) <= 2e-243:
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m)
	elif (l * l) <= 1e+254:
		tmp = ((2.0 * math.cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m)
	else:
		tmp = 2.0 / (t * (((t_1 * k_m) / l) * (k_m / (math.cos(k_m) * l))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 2e-243)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(k_m / l) * k_m)) / Float64(l / k_m)) * k_m));
	elseif (Float64(l * l) <= 1e+254)
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(t_1 * t) * k_m)) * Float64(Float64(l * l) / k_m));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(t_1 * k_m) / l) * Float64(k_m / Float64(cos(k_m) * l)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if ((l * l) <= 2e-243)
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	elseif ((l * l) <= 1e+254)
		tmp = ((2.0 * cos(k_m)) / ((t_1 * t) * k_m)) * ((l * l) / k_m);
	else
		tmp = 2.0 / (t * (((t_1 * k_m) / l) * (k_m / (cos(k_m) * l))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 2e-243], N[(2.0 / N[(N[(N[(t * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+254], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(t$95$1 * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.99999999999999999e-243

   1. Initial program 23.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 81.3%

      \[\leadsto \frac{2}{\frac{{\ell}^{-1} \cdot {\left({k}^{-4}\right)}^{-1}}{\ell} \cdot t} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.9%

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

   11. Applied rewrites 96.5%

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

   if 1.99999999999999999e-243 < (*.f64 l l) < 9.9999999999999994e253

   1. Initial program 44.1%

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

   3. Taylor expanded in 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 99.0%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 9.9999999999999994e253 < (*.f64 l l)

   1. Initial program 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot k\right)}{\frac{\ell}{k}} \cdot k}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+254}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left({\sin k}^{2} \cdot t\right) \cdot k} \cdot \frac{\ell \cdot \ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{{\sin k}^{2} \cdot k}{\ell} \cdot \frac{k}{\cos k \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t}{\frac{\frac{\ell}{k\_m}}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k\_m}{\cos k\_m}}{\ell} \cdot \left(t \cdot {\sin k\_m}^{2}\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.8e-56)
   (/ 2.0 (/ (* (* (/ k_m l) k_m) t) (/ (/ l k_m) k_m)))
   (/
    2.0
    (* (* (/ (/ k_m (cos k_m)) l) (* t (pow (sin k_m) 2.0))) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.8e-56) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / ((((k_m / cos(k_m)) / l) * (t * pow(sin(k_m), 2.0))) * (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) :: tmp
    if (k_m <= 4.8d-56) then
        tmp = 2.0d0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
    else
        tmp = 2.0d0 / ((((k_m / cos(k_m)) / l) * (t * (sin(k_m) ** 2.0d0))) * (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 tmp;
	if (k_m <= 4.8e-56) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / ((((k_m / Math.cos(k_m)) / l) * (t * Math.pow(Math.sin(k_m), 2.0))) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.8e-56:
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
	else:
		tmp = 2.0 / ((((k_m / math.cos(k_m)) / l) * (t * math.pow(math.sin(k_m), 2.0))) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.8e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / l) * k_m) * t) / Float64(Float64(l / k_m) / k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / cos(k_m)) / l) * Float64(t * (sin(k_m) ^ 2.0))) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.8e-56)
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	else
		tmp = 2.0 / ((((k_m / cos(k_m)) / l) * (t * (sin(k_m) ^ 2.0))) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.8e-56], N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.80000000000000001e-56

   1. Initial program 41.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 86.4%

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

   if 4.80000000000000001e-56 < k

   1. Initial program 32.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 95.2%

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

   7. Applied rewrites 99.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t}{\frac{\frac{\ell}{k\_m}}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)}{\ell \cdot \frac{\frac{\ell}{k\_m}}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.8e-56)
   (/ 2.0 (/ (* (* (/ k_m l) k_m) t) (/ (/ l k_m) k_m)))
   (/ 2.0 (/ (* k_m (* (sin k_m) (tan k_m))) (* l (/ (/ l k_m) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.8e-56) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / ((k_m * (sin(k_m) * tan(k_m))) / (l * ((l / k_m) / t)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.8d-56) then
        tmp = 2.0d0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
    else
        tmp = 2.0d0 / ((k_m * (sin(k_m) * tan(k_m))) / (l * ((l / k_m) / t)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.8e-56) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / ((k_m * (Math.sin(k_m) * Math.tan(k_m))) / (l * ((l / k_m) / t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.8e-56:
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
	else:
		tmp = 2.0 / ((k_m * (math.sin(k_m) * math.tan(k_m))) / (l * ((l / k_m) / t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.8e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / l) * k_m) * t) / Float64(Float64(l / k_m) / k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(sin(k_m) * tan(k_m))) / Float64(l * Float64(Float64(l / k_m) / t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.8e-56)
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	else
		tmp = 2.0 / ((k_m * (sin(k_m) * tan(k_m))) / (l * ((l / k_m) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.8e-56], N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.80000000000000001e-56

   1. Initial program 41.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 86.4%

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

   if 4.80000000000000001e-56 < k

   1. Initial program 32.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 95.2%

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 90.9%

      \[\leadsto \frac{2}{\frac{\left(k \cdot \left(\sin k \cdot \tan k\right)\right) \cdot 1}{\color{blue}{\ell \cdot \frac{\frac{\ell}{k}}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \frac{\frac{\ell}{k}}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\frac{\ell}{k\_m}} \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1e-70)
   (/ 2.0 (* (/ (* t (* (/ k_m l) k_m)) (/ l k_m)) k_m))
   (/ 2.0 (* (* k_m (* (sin k_m) (tan k_m))) (* (/ t l) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-70) {
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	} else {
		tmp = 2.0 / ((k_m * (sin(k_m) * tan(k_m))) * ((t / l) * (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) :: tmp
    if (k_m <= 1d-70) then
        tmp = 2.0d0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m)
    else
        tmp = 2.0d0 / ((k_m * (sin(k_m) * tan(k_m))) * ((t / l) * (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 tmp;
	if (k_m <= 1e-70) {
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	} else {
		tmp = 2.0 / ((k_m * (Math.sin(k_m) * Math.tan(k_m))) * ((t / l) * (k_m / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1e-70:
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m)
	else:
		tmp = 2.0 / ((k_m * (math.sin(k_m) * math.tan(k_m))) * ((t / l) * (k_m / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1e-70)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(k_m / l) * k_m)) / Float64(l / k_m)) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(sin(k_m) * tan(k_m))) * Float64(Float64(t / l) * Float64(k_m / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1e-70)
		tmp = 2.0 / (((t * ((k_m / l) * k_m)) / (l / k_m)) * k_m);
	else
		tmp = 2.0 / ((k_m * (sin(k_m) * tan(k_m))) * ((t / l) * (k_m / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1e-70], N[(2.0 / N[(N[(N[(t * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999996e-71

   1. Initial program 41.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.7%

      \[\leadsto \frac{2}{\frac{{\ell}^{-1} \cdot {\left({k}^{-4}\right)}^{-1}}{\ell} \cdot t} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.9%

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

   11. Applied rewrites 86.7%

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

   if 9.99999999999999996e-71 < k

   1. Initial program 33.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 94.0%

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

   7. Applied rewrites 98.9%

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

   9. Applied rewrites 91.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot k\right)}{\frac{\ell}{k}} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t}{\frac{\frac{\ell}{k\_m}}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot t\right) \cdot \left(k\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.8e-56)
   (/ 2.0 (/ (* (* (/ k_m l) k_m) t) (/ (/ l k_m) k_m)))
   (/ 2.0 (/ (* (* k_m t) (* k_m (* (sin k_m) (tan k_m)))) (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.8e-56) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / (((k_m * t) * (k_m * (sin(k_m) * tan(k_m)))) / (l * 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) :: tmp
    if (k_m <= 4.8d-56) then
        tmp = 2.0d0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
    else
        tmp = 2.0d0 / (((k_m * t) * (k_m * (sin(k_m) * tan(k_m)))) / (l * 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 tmp;
	if (k_m <= 4.8e-56) {
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	} else {
		tmp = 2.0 / (((k_m * t) * (k_m * (Math.sin(k_m) * Math.tan(k_m)))) / (l * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.8e-56:
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m))
	else:
		tmp = 2.0 / (((k_m * t) * (k_m * (math.sin(k_m) * math.tan(k_m)))) / (l * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.8e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / l) * k_m) * t) / Float64(Float64(l / k_m) / k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * t) * Float64(k_m * Float64(sin(k_m) * tan(k_m)))) / Float64(l * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.8e-56)
		tmp = 2.0 / ((((k_m / l) * k_m) * t) / ((l / k_m) / k_m));
	else
		tmp = 2.0 / (((k_m * t) * (k_m * (sin(k_m) * tan(k_m)))) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.8e-56], N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.80000000000000001e-56

   1. Initial program 41.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 86.4%

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

   if 4.80000000000000001e-56 < k

   1. Initial program 32.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 95.2%

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 74.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot \left(k \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 71.9

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

5. Applied rewrites 71.9%

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

7. Applied rewrites 71.9%

   \[\leadsto \frac{2}{\frac{{\ell}^{-1} \cdot {\left({k}^{-4}\right)}^{-1}}{\ell} \cdot t} \]
8. Step-by-step derivation

9. Applied rewrites 65.2%

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

11. Applied rewrites 78.9%

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

12. Final simplification 78.9%

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

Alternative 9: 75.9% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / (k_m * ((k_m / l) * (((k_m / l) * k_m) * t)))
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 / l) * (((k_m / l) * k_m) * t)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 71.9

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

5. Applied rewrites 71.9%

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

7. Applied rewrites 74.6%

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

9. Applied rewrites 78.8%

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

Alternative 10: 64.1% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / (((k_m * k_m) * ((k_m * k_m) / (l * l))) * t)
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) * ((k_m * k_m) / (l * l))) * t);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 71.9

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

5. Applied rewrites 71.9%

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

7. Applied rewrites 71.9%

   \[\leadsto \frac{2}{\frac{{\ell}^{-1} \cdot {\left({k}^{-4}\right)}^{-1}}{\ell} \cdot t} \]
8. Step-by-step derivation

9. Applied rewrites 65.2%

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

11. Applied rewrites 67.0%

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

Reproduce

herbie shell --seed 2024307 
(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))))