Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 98.3%

Time: 15.5s

Alternatives: 20

Speedup: 9.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.8% 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: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.22 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k\_m}^{2} \cdot \left(t \cdot \frac{k\_m}{\cos k\_m \cdot \ell}\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.22e-140)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (/
    2.0
    (* (* (pow (sin k_m) 2.0) (* t (/ k_m (* (cos k_m) l)))) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.22e-140) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((pow(sin(k_m), 2.0) * (t * (k_m / (cos(k_m) * 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 <= 1.22d-140) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    else
        tmp = 2.0d0 / (((sin(k_m) ** 2.0d0) * (t * (k_m / (cos(k_m) * 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 <= 1.22e-140) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k_m), 2.0) * (t * (k_m / (Math.cos(k_m) * l)))) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.22e-140:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	else:
		tmp = 2.0 / ((math.pow(math.sin(k_m), 2.0) * (t * (k_m / (math.cos(k_m) * l)))) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.22e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k_m) ^ 2.0) * Float64(t * Float64(k_m / Float64(cos(k_m) * 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 <= 1.22e-140)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	else
		tmp = 2.0 / (((sin(k_m) ^ 2.0) * (t * (k_m / (cos(k_m) * 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, 1.22e-140], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.22e-140

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 79.9%

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

   if 1.22e-140 < k

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

   11. Applied rewrites 98.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \left(t \cdot \frac{k}{\cos k \cdot \ell}\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2} \cdot t\\ \mathbf{if}\;k\_m \leq 0.00032:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k\_m}^{3}, k\_m\right) \cdot \left(\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t\right)\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 4.9 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_1 \cdot k\_m\right) \cdot k\_m}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot t\_1\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (pow (sin k_m) 2.0) t)))
   (if (<= k_m 0.00032)
     (/
      2.0
      (*
       (* (fma -0.3333333333333333 (pow k_m 3.0) k_m) (* (* (/ k_m l) k_m) t))
       (/ (/ k_m (cos k_m)) l)))
     (if (<= k_m 4.9e+187)
       (/ 2.0 (/ (* (* t_1 k_m) k_m) (* (* (cos k_m) l) l)))
       (/ 2.0 (* (* (/ k_m l) t_1) (/ 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) * t;
	double tmp;
	if (k_m <= 0.00032) {
		tmp = 2.0 / ((fma(-0.3333333333333333, pow(k_m, 3.0), k_m) * (((k_m / l) * k_m) * t)) * ((k_m / cos(k_m)) / l));
	} else if (k_m <= 4.9e+187) {
		tmp = 2.0 / (((t_1 * k_m) * k_m) / ((cos(k_m) * l) * l));
	} else {
		tmp = 2.0 / (((k_m / l) * t_1) * (k_m / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64((sin(k_m) ^ 2.0) * t)
	tmp = 0.0
	if (k_m <= 0.00032)
		tmp = Float64(2.0 / Float64(Float64(fma(-0.3333333333333333, (k_m ^ 3.0), k_m) * Float64(Float64(Float64(k_m / l) * k_m) * t)) * Float64(Float64(k_m / cos(k_m)) / l)));
	elseif (k_m <= 4.9e+187)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * k_m) * k_m) / Float64(Float64(cos(k_m) * l) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * t_1) * Float64(k_m / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k$95$m, 0.00032], N[(2.0 / N[(N[(N[(-0.3333333333333333 * N[Power[k$95$m, 3.0], $MachinePrecision] + k$95$m), $MachinePrecision] * N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.9e+187], N[(2.0 / N[(N[(N[(t$95$1 * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.20000000000000026e-4

   1. Initial program 37.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 74.2%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{3}}\right)} \]

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 83.2%

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right)}\right)} \]

   if 3.20000000000000026e-4 < k < 4.9000000000000003e187

   1. Initial program 19.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 83.0%

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

   if 4.9000000000000003e187 < k

   1. Initial program 46.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 75.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00032:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 0.00032:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k\_m}^{3}, k\_m\right) \cdot \left(\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t\right)\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot \frac{\cos k\_m \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \left(t\_1 \cdot t\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= k_m 0.00032)
     (/
      2.0
      (*
       (* (fma -0.3333333333333333 (pow k_m 3.0) k_m) (* (* (/ k_m l) k_m) t))
       (/ (/ k_m (cos k_m)) l)))
     (if (<= k_m 8e+150)
       (* (/ (* l l) t) (/ (* (cos k_m) 2.0) (* (* k_m k_m) t_1)))
       (/ 2.0 (* (* (/ k_m l) (* t_1 t)) (/ 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 (k_m <= 0.00032) {
		tmp = 2.0 / ((fma(-0.3333333333333333, pow(k_m, 3.0), k_m) * (((k_m / l) * k_m) * t)) * ((k_m / cos(k_m)) / l));
	} else if (k_m <= 8e+150) {
		tmp = ((l * l) / t) * ((cos(k_m) * 2.0) / ((k_m * k_m) * t_1));
	} else {
		tmp = 2.0 / (((k_m / l) * (t_1 * t)) * (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 (k_m <= 0.00032)
		tmp = Float64(2.0 / Float64(Float64(fma(-0.3333333333333333, (k_m ^ 3.0), k_m) * Float64(Float64(Float64(k_m / l) * k_m) * t)) * Float64(Float64(k_m / cos(k_m)) / l)));
	elseif (k_m <= 8e+150)
		tmp = Float64(Float64(Float64(l * l) / t) * Float64(Float64(cos(k_m) * 2.0) / Float64(Float64(k_m * k_m) * t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(t_1 * t)) * Float64(k_m / l)));
	end
	return 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[k$95$m, 0.00032], N[(2.0 / N[(N[(N[(-0.3333333333333333 * N[Power[k$95$m, 3.0], $MachinePrecision] + k$95$m), $MachinePrecision] * N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8e+150], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.20000000000000026e-4

   1. Initial program 37.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 74.2%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{3}}\right)} \]

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 83.2%

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right)}\right)} \]

   if 3.20000000000000026e-4 < k < 7.99999999999999985e150

   1. Initial program 22.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 95.0%

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

   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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 88.7%

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

   if 7.99999999999999985e150 < k

   1. Initial program 37.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 90.6%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 67.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00032:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot \frac{\cos k \cdot 2}{\left(k \cdot k\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k\_m}^{2} \cdot t\right) \cdot \frac{k\_m}{\cos k\_m \cdot \ell}\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e-140)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (/
    2.0
    (* (* (* (pow (sin k_m) 2.0) t) (/ k_m (* (cos k_m) l))) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / (((pow(sin(k_m), 2.0) * t) * (k_m / (cos(k_m) * 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 <= 2.65d-140) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    else
        tmp = 2.0d0 / ((((sin(k_m) ** 2.0d0) * t) * (k_m / (cos(k_m) * 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 <= 2.65e-140) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / (((Math.pow(Math.sin(k_m), 2.0) * t) * (k_m / (Math.cos(k_m) * l))) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e-140:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	else:
		tmp = 2.0 / (((math.pow(math.sin(k_m), 2.0) * t) * (k_m / (math.cos(k_m) * l))) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) * Float64(k_m / Float64(cos(k_m) * 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 <= 2.65e-140)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	else
		tmp = 2.0 / ((((sin(k_m) ^ 2.0) * t) * (k_m / (cos(k_m) * 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, 2.65e-140], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.64999999999999992e-140

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 79.9%

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

   if 2.64999999999999992e-140 < k

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k}^{2} \cdot t\right) \cdot \frac{k}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k\_m}^{2} \cdot t}{\cos k\_m \cdot \ell} \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e-140)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (/
    2.0
    (* (* (/ (* (pow (sin k_m) 2.0) t) (* (cos k_m) l)) k_m) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((((pow(sin(k_m), 2.0) * t) / (cos(k_m) * l)) * k_m) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.65d-140) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    else
        tmp = 2.0d0 / (((((sin(k_m) ** 2.0d0) * t) / (cos(k_m) * l)) * k_m) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((((Math.pow(Math.sin(k_m), 2.0) * t) / (Math.cos(k_m) * l)) * k_m) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e-140:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	else:
		tmp = 2.0 / ((((math.pow(math.sin(k_m), 2.0) * t) / (math.cos(k_m) * l)) * k_m) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) / Float64(cos(k_m) * l)) * k_m) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.65e-140)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	else
		tmp = 2.0 / (((((sin(k_m) ^ 2.0) * t) / (cos(k_m) * l)) * k_m) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.65e-140], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.64999999999999992e-140

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 79.9%

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

   if 2.64999999999999992e-140 < k

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

   11. Applied rewrites 96.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\cos k \cdot \ell} \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 / l) * (((sin(k_m) * t) * sin(k_m)) * (k_m / (cos(k_m) * l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 93.2%

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

7. Applied rewrites 95.4%

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

9. Applied rewrites 95.5%

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

11. Applied rewrites 97.0%

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

12. Final simplification 97.0%

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

Alternative 7: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 185000:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k\_m}^{3}, k\_m\right) \cdot \left(\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t\right)\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 4 \cdot 10^{+108}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{t} \cdot \frac{k\_m}{t}\right) \cdot \left(\tan k\_m \cdot \left(\left(\frac{\frac{t}{\ell} \cdot t}{\ell} \cdot t\right) \cdot \sin k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \left({\sin k\_m}^{2} \cdot t\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 185000.0)
   (/
    2.0
    (*
     (* (fma -0.3333333333333333 (pow k_m 3.0) k_m) (* (* (/ k_m l) k_m) t))
     (/ (/ k_m (cos k_m)) l)))
   (if (<= k_m 4e+108)
     (/
      2.0
      (*
       (* (/ k_m t) (/ k_m t))
       (* (tan k_m) (* (* (/ (* (/ t l) t) l) t) (sin k_m)))))
     (/ 2.0 (* (* (/ k_m l) (* (pow (sin k_m) 2.0) t)) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 185000.0) {
		tmp = 2.0 / ((fma(-0.3333333333333333, pow(k_m, 3.0), k_m) * (((k_m / l) * k_m) * t)) * ((k_m / cos(k_m)) / l));
	} else if (k_m <= 4e+108) {
		tmp = 2.0 / (((k_m / t) * (k_m / t)) * (tan(k_m) * (((((t / l) * t) / l) * t) * sin(k_m))));
	} else {
		tmp = 2.0 / (((k_m / l) * (pow(sin(k_m), 2.0) * t)) * (k_m / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 185000.0)
		tmp = Float64(2.0 / Float64(Float64(fma(-0.3333333333333333, (k_m ^ 3.0), k_m) * Float64(Float64(Float64(k_m / l) * k_m) * t)) * Float64(Float64(k_m / cos(k_m)) / l)));
	elseif (k_m <= 4e+108)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / t) * Float64(k_m / t)) * Float64(tan(k_m) * Float64(Float64(Float64(Float64(Float64(t / l) * t) / l) * t) * sin(k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64((sin(k_m) ^ 2.0) * t)) * Float64(k_m / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 185000.0], N[(2.0 / N[(N[(N[(-0.3333333333333333 * N[Power[k$95$m, 3.0], $MachinePrecision] + k$95$m), $MachinePrecision] * N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4e+108], N[(2.0 / N[(N[(N[(k$95$m / t), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 185000

   1. Initial program 37.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 93.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 73.8%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{3}}\right)} \]

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 82.8%

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right)}\right)} \]

   if 185000 < k < 4.0000000000000001e108

   1. Initial program 29.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 46.3

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 80.7

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

   7. Applied rewrites 80.7%

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

   if 4.0000000000000001e108 < k

   1. Initial program 31.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 67.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 185000:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+108}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\tan k \cdot \left(\left(\frac{\frac{t}{\ell} \cdot t}{\ell} \cdot t\right) \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot k\_m}{t \cdot t} \cdot \left(\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k\_m}^{3}, k\_m\right) \cdot \left(\left(\frac{k\_m}{\ell} \cdot k\_m\right) \cdot t\right)\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.7e-146)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (if (<= t 5.1e+95)
     (/
      2.0
      (*
       (/ (* k_m k_m) (* t t))
       (* (* (* (tan k_m) (sin k_m)) (/ t l)) (/ (* t t) l))))
     (/
      2.0
      (*
       (* (fma -0.3333333333333333 (pow k_m 3.0) k_m) (* (* (/ k_m l) k_m) t))
       (/ (/ k_m (cos k_m)) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.7e-146) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else if (t <= 5.1e+95) {
		tmp = 2.0 / (((k_m * k_m) / (t * t)) * (((tan(k_m) * sin(k_m)) * (t / l)) * ((t * t) / l)));
	} else {
		tmp = 2.0 / ((fma(-0.3333333333333333, pow(k_m, 3.0), k_m) * (((k_m / l) * k_m) * t)) * ((k_m / cos(k_m)) / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.7e-146)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	elseif (t <= 5.1e+95)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) / Float64(t * t)) * Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(t / l)) * Float64(Float64(t * t) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(-0.3333333333333333, (k_m ^ 3.0), k_m) * Float64(Float64(Float64(k_m / l) * k_m) * t)) * Float64(Float64(k_m / cos(k_m)) / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.7e-146], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.1e+95], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(-0.3333333333333333 * N[Power[k$95$m, 3.0], $MachinePrecision] + k$95$m), $MachinePrecision] * N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.69999999999999995e-146

   1. Initial program 32.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 60.9%

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

   9. Applied rewrites 76.1%

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

   if 2.69999999999999995e-146 < t < 5.10000000000000003e95

   1. Initial program 63.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. metadata-eval N/A

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

      6. +-rgt-identity 69.5

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 84.1

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

   8. Applied rewrites 84.1%

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

   if 5.10000000000000003e95 < t

   1. Initial program 16.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 83.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 66.0%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{3}}\right)} \]

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 85.5%

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{t \cdot t} \cdot \left(\left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.3333333333333333, {k}^{3}, k\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \left({\sin k\_m}^{2} \cdot t\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e-140)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (/ 2.0 (* (* (/ k_m l) (* (pow (sin k_m) 2.0) t)) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / (((k_m / l) * (pow(sin(k_m), 2.0) * t)) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.65d-140) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    else
        tmp = 2.0d0 / (((k_m / l) * ((sin(k_m) ** 2.0d0) * t)) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / (((k_m / l) * (Math.pow(Math.sin(k_m), 2.0) * t)) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e-140:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	else:
		tmp = 2.0 / (((k_m / l) * (math.pow(math.sin(k_m), 2.0) * t)) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64((sin(k_m) ^ 2.0) * t)) * 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 <= 2.65e-140)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	else
		tmp = 2.0 / (((k_m / l) * ((sin(k_m) ^ 2.0) * t)) * (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, 2.65e-140], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.64999999999999992e-140

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 79.9%

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

   if 2.64999999999999992e-140 < k

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= l 2.2e-220)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ 2.0 (* (/ (* (* (pow (sin k_m) 2.0) t) k_m) l) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (l <= 2.2e-220) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((((pow(sin(k_m), 2.0) * t) * k_m) / 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) :: t_1
    real(8) :: tmp
    t_1 = (k_m / l) * k_m
    if (l <= 2.2d-220) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / (((((sin(k_m) ** 2.0d0) * t) * k_m) / 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 t_1 = (k_m / l) * k_m;
	double tmp;
	if (l <= 2.2e-220) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((((Math.pow(Math.sin(k_m), 2.0) * t) * k_m) / l) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if l <= 2.2e-220:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / ((((math.pow(math.sin(k_m), 2.0) * t) * k_m) / l) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (l <= 2.2e-220)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m) / l) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (l <= 2.2e-220)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / (((((sin(k_m) ^ 2.0) * t) * k_m) / l) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[l, 2.2e-220], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.19999999999999987e-220

   1. Initial program 34.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 61.8%

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

   9. Applied rewrites 76.7%

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

   if 2.19999999999999987e-220 < l

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 76.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-209}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.4e-209)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (/ 2.0 (* (* (* (* t k_m) k_m) (/ (/ k_m (cos k_m)) l)) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.4e-209) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((((t * k_m) * k_m) * ((k_m / cos(k_m)) / 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 (t <= 4.4d-209) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    else
        tmp = 2.0d0 / ((((t * k_m) * k_m) * ((k_m / cos(k_m)) / 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 (t <= 4.4e-209) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((((t * k_m) * k_m) * ((k_m / Math.cos(k_m)) / l)) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4.4e-209:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	else:
		tmp = 2.0 / ((((t * k_m) * k_m) * ((k_m / math.cos(k_m)) / l)) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.4e-209)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * k_m) * k_m) * Float64(Float64(k_m / cos(k_m)) / 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 (t <= 4.4e-209)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	else
		tmp = 2.0 / ((((t * k_m) * k_m) * ((k_m / cos(k_m)) / l)) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.4e-209], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.40000000000000019e-209

   1. Initial program 33.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 60.2%

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

   9. Applied rewrites 77.3%

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

   if 4.40000000000000019e-209 < t

   1. Initial program 37.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 90.4%

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 74.3%

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

   12. Applied rewrites 76.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-209}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\frac{k}{\cos k}}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e-140)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (/ 2.0 (* (* (* (* k_m k_m) t) (/ (/ k_m (cos k_m)) l)) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) * ((k_m / cos(k_m)) / 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 <= 2.65d-140) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    else
        tmp = 2.0d0 / ((((k_m * k_m) * t) * ((k_m / cos(k_m)) / 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 <= 2.65e-140) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) * ((k_m / Math.cos(k_m)) / l)) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e-140:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	else:
		tmp = 2.0 / ((((k_m * k_m) * t) * ((k_m / math.cos(k_m)) / l)) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m / cos(k_m)) / 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 <= 2.65e-140)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	else
		tmp = 2.0 / ((((k_m * k_m) * t) * ((k_m / cos(k_m)) / 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, 2.65e-140], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.64999999999999992e-140

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 79.9%

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

   if 2.64999999999999992e-140 < k

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 69.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{k}{\cos k}}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot k\_m}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-140)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))
   (/ 2.0 (/ (* (* (* (* k_m k_m) t) (/ k_m l)) k_m) (* (cos k_m) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-140) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (cos(k_m) * l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.2d-140) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    else
        tmp = 2.0d0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (cos(k_m) * l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-140) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (Math.cos(k_m) * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.2e-140:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	else:
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (math.cos(k_m) * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m / l)) * k_m) / Float64(cos(k_m) * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.2e-140)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	else
		tmp = 2.0 / (((((k_m * k_m) * t) * (k_m / l)) * k_m) / (cos(k_m) * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-140], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000001e-140

   1. Initial program 36.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 79.9%

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

   if 3.2000000000000001e-140 < k

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 69.0%

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

   12. Applied rewrites 69.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot k}{\cos k \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;k\_m \leq 0.00034:\\ \;\;\;\;\frac{2}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot \frac{k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) t) k_m) k_m))))
   (if (<= k_m 2.65e-140)
     t_1
     (if (<= k_m 0.00034)
       (/ 2.0 (* (* (* (* k_m k_m) t) (/ k_m l)) (/ k_m l)))
       t_1))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 2.0 / (((pow((k_m / l), 2.0) * t) * k_m) * k_m);
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = t_1;
	} else if (k_m <= 0.00034) {
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l));
	} else {
		tmp = t_1;
	}
	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 = 2.0d0 / (((((k_m / l) ** 2.0d0) * t) * k_m) * k_m)
    if (k_m <= 2.65d-140) then
        tmp = t_1
    else if (k_m <= 0.00034d0) then
        tmp = 2.0d0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 2.0 / (((Math.pow((k_m / l), 2.0) * t) * k_m) * k_m);
	double tmp;
	if (k_m <= 2.65e-140) {
		tmp = t_1;
	} else if (k_m <= 0.00034) {
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = 2.0 / (((math.pow((k_m / l), 2.0) * t) * k_m) * k_m)
	tmp = 0
	if k_m <= 2.65e-140:
		tmp = t_1
	elif k_m <= 0.00034:
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l))
	else:
		tmp = t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * t) * k_m) * k_m))
	tmp = 0.0
	if (k_m <= 2.65e-140)
		tmp = t_1;
	elseif (k_m <= 0.00034)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m / l)) * Float64(k_m / l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = 2.0 / (((((k_m / l) ^ 2.0) * t) * k_m) * k_m);
	tmp = 0.0;
	if (k_m <= 2.65e-140)
		tmp = t_1;
	elseif (k_m <= 0.00034)
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2.65e-140], t$95$1, If[LessEqual[k$95$m, 0.00034], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.64999999999999992e-140 or 3.4e-4 < k

   1. Initial program 34.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 73.7%

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

   if 2.64999999999999992e-140 < k < 3.4e-4

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 99.7%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 98.3%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 98.4%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;k \leq 0.00034:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \leq 1.75 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k\_m \cdot k\_m, 1\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= l 1.75e-31)
     (/ 2.0 (* (* t_1 t_1) t))
     (/
      2.0
      (*
       (*
        (* (/ (fma 0.16666666666666666 (* k_m k_m) 1.0) l) (/ t l))
        (* k_m k_m))
       (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (l <= 1.75e-31) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((((fma(0.16666666666666666, (k_m * k_m), 1.0) / l) * (t / l)) * (k_m * k_m)) * (k_m * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (l <= 1.75e-31)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(0.16666666666666666, Float64(k_m * k_m), 1.0) / l) * Float64(t / l)) * Float64(k_m * k_m)) * Float64(k_m * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[l, 1.75e-31], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.74999999999999993e-31

   1. Initial program 33.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 78.4%

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

   if 1.74999999999999993e-31 < l

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. metadata-eval N/A

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

      6. +-rgt-identity 44.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. lower-/.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 24.8

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

   4. Applied rewrites 24.8%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{t}{{\ell}^{2}}\right) \cdot {k}^{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{1}{6}} + \frac{t}{{\ell}^{2}}\right) \cdot {k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)} \cdot \frac{1}{6} + \frac{t}{{\ell}^{2}}\right) \cdot {k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{1}{6}\right)} + \frac{t}{{\ell}^{2}}\right) \cdot {k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)} + \frac{t}{{\ell}^{2}}\right) \cdot {k}^{4}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{1}{6}\right)} + \frac{t}{{\ell}^{2}}\right) \cdot {k}^{4}} \]

      7. metadata-eval N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 60.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(k \cdot k, 0.16666666666666666, 1\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot {k}^{4}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 75.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-132}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\ell}}{\ell} \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= (* l l) 1e-132)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ 2.0 (* (/ (/ (* (* k_m k_m) t) l) l) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if ((l * l) <= 1e-132) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) / l) / l) * (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k_m / l) * k_m
    if ((l * l) <= 1d-132) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / (((((k_m * k_m) * t) / l) / l) * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if ((l * l) <= 1e-132) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) / l) / l) * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if (l * l) <= 1e-132:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / (((((k_m * k_m) * t) / l) / l) * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (Float64(l * l) <= 1e-132)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) / l) / l) * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if ((l * l) <= 1e-132)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / (((((k_m * k_m) * t) / l) / l) * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-132], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999999e-133

   1. Initial program 22.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 65.2%

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

   9. Applied rewrites 92.5%

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

   if 9.9999999999999999e-133 < (*.f64 l l)

   1. Initial program 43.5%

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

   3. Taylor expanded in 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 62.3

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

   5. Applied rewrites 62.3%

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

   7. Applied rewrites 60.1%

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

   9. Applied rewrites 62.2%

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

   11. Applied rewrites 66.7%

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

Alternative 17: 74.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;t \leq 8 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= t 8e+186)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ 2.0 (* (* (* (* k_m k_m) t) (/ k_m l)) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (t <= 8e+186) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / 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) :: t_1
    real(8) :: tmp
    t_1 = (k_m / l) * k_m
    if (t <= 8d+186) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / ((((k_m * k_m) * t) * (k_m / 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 t_1 = (k_m / l) * k_m;
	double tmp;
	if (t <= 8e+186) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if t <= 8e+186:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (t <= 8e+186)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m / l)) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if (t <= 8e+186)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / ((((k_m * k_m) * t) * (k_m / l)) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[t, 8e+186], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.99999999999999984e186

   1. Initial program 37.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 62.1%

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

   9. Applied rewrites 75.6%

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

   if 7.99999999999999984e186 < t

   1. Initial program 17.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 81.6%

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 74.1%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 73.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.9% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 93.2%

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

7. Applied rewrites 95.4%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 75.0%

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

11. Taylor expanded in k around 0

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

13. Applied rewrites 74.0%

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

14. Final simplification 74.0%

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

Alternative 19: 64.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 68.7

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

5. Applied rewrites 68.7%

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

7. Applied rewrites 62.0%

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

9. Applied rewrites 62.4%

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

11. Applied rewrites 63.0%

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

Alternative 20: 65.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 68.7

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

5. Applied rewrites 68.7%

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

7. Applied rewrites 62.0%

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

9. Applied rewrites 61.0%

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

11. Applied rewrites 62.7%

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

12. Final simplification 62.7%

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

Reproduce

herbie shell --seed 2024273 
(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))))