Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 98.2%

Time: 15.6s

Alternatives: 17

Speedup: 11.0×

• 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: 35.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \left(\left(t \cdot k\_m\right) \cdot \left(\sin k\_m \cdot \frac{\tan k\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot \frac{k\_m}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.4e-6)
   (/ 2.0 (* (/ k_m l) (* (* t k_m) (* (sin k_m) (/ (tan k_m) l)))))
   (/
    2.0
    (*
     (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) 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 <= 4.4e-6) {
		tmp = 2.0 / ((k_m / l) * ((t * k_m) * (sin(k_m) * (tan(k_m) / l))));
	} else {
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * 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 <= 4.4d-6) then
        tmp = 2.0d0 / ((k_m / l) * ((t * k_m) * (sin(k_m) * (tan(k_m) / l))))
    else
        tmp = 2.0d0 / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * 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 <= 4.4e-6) {
		tmp = 2.0 / ((k_m / l) * ((t * k_m) * (Math.sin(k_m) * (Math.tan(k_m) / l))));
	} else {
		tmp = 2.0 / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * 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 <= 4.4e-6:
		tmp = 2.0 / ((k_m / l) * ((t * k_m) * (math.sin(k_m) * (math.tan(k_m) / l))))
	else:
		tmp = 2.0 / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * 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 <= 4.4e-6)
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64(Float64(t * k_m) * Float64(sin(k_m) * Float64(tan(k_m) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * Float64(k_m / l)) * Float64(k_m / Float64(cos(k_m) * l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.4e-6)
		tmp = 2.0 / ((k_m / l) * ((t * k_m) * (sin(k_m) * (tan(k_m) / l))));
	else
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * 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, 4.4e-6], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(t * k$95$m), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4000000000000002e-6

   1. Initial program 39.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 71.0

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 73.4%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}{\cos k \cdot \ell}}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.7%

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

   11. Applied rewrites 95.5%

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

   if 4.4000000000000002e-6 < k

   1. Initial program 24.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 99.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot k\right) \cdot \left(\sin k \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\cos k \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \left(\left(t \cdot k\_m\right) \cdot \left(\sin k\_m \cdot \frac{\tan k\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.5, \cos \left(k\_m + k\_m\right), 0.5\right) \cdot \left(\frac{k\_m}{\ell} \cdot t\right)\right) \cdot \frac{k\_m}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e+68)
   (/ 2.0 (* (/ k_m l) (* (* t k_m) (* (sin k_m) (/ (tan k_m) l)))))
   (/
    2.0
    (*
     (* (fma -0.5 (cos (+ k_m k_m)) 0.5) (* (/ k_m l) t))
     (/ k_m (* (cos k_m) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e+68) {
		tmp = 2.0 / ((k_m / l) * ((t * k_m) * (sin(k_m) * (tan(k_m) / l))));
	} else {
		tmp = 2.0 / ((fma(-0.5, cos((k_m + k_m)), 0.5) * ((k_m / l) * t)) * (k_m / (cos(k_m) * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e+68)
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64(Float64(t * k_m) * Float64(sin(k_m) * Float64(tan(k_m) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(-0.5, cos(Float64(k_m + k_m)), 0.5) * Float64(Float64(k_m / l) * t)) * Float64(k_m / Float64(cos(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, 7.5e+68], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(t * k$95$m), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(-0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999959e68

   1. Initial program 38.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 75.1%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}{\cos k \cdot \ell}}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.2%

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

   11. Applied rewrites 95.8%

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

   if 7.49999999999999959e68 < k

   1. Initial program 27.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 95.7%

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

   9. Applied rewrites 99.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot k\right) \cdot \left(\sin k \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.5, \cos \left(k + k\right), 0.5\right) \cdot \left(\frac{k}{\ell} \cdot t\right)\right) \cdot \frac{k}{\cos k \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.8 \cdot 10^{+182}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \left(\left(t \cdot k\_m\right) \cdot \left(\sin k\_m \cdot \frac{\tan k\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{k\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.8e+182)
   (/ 2.0 (* (/ k_m l) (* (* t k_m) (* (sin k_m) (/ (tan k_m) l)))))
   (/
    2.0
    (*
     (* (fma (cos (+ k_m k_m)) -0.5 0.5) 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 <= 2.8e+182) {
		tmp = 2.0 / ((k_m / l) * ((t * k_m) * (sin(k_m) * (tan(k_m) / l))));
	} else {
		tmp = 2.0 / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * ((k_m / (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 <= 2.8e+182)
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64(Float64(t * k_m) * Float64(sin(k_m) * Float64(tan(k_m) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * Float64(Float64(k_m / Float64(cos(k_m) * l)) * 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, 2.8e+182], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(t * k$95$m), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.80000000000000006e182

   1. Initial program 35.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 71.8

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

   5. Applied rewrites 71.8%

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

   7. Applied rewrites 76.2%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}{\cos k \cdot \ell}}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.5%

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

   11. Applied rewrites 95.7%

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

   if 2.80000000000000006e182 < k

   1. Initial program 44.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{+182}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot k\right) \cdot \left(\sin k \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;k\_m \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k\_m \cdot k\_m, 1\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{elif}\;k\_m \leq 2.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{\ell}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\ell \cdot 2} \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_1 \cdot \left(t \cdot k\_m\right)}{\ell \cdot \ell} \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= k_m 3e-69)
     (/
      2.0
      (*
       (* (* (* (fma -0.3333333333333333 (* k_m k_m) 1.0) t) k_m) k_m)
       (* (/ k_m (* (cos k_m) l)) (/ k_m l))))
     (if (<= k_m 2.1e+115)
       (/ l (* (/ (* (* k_m k_m) t) (* l 2.0)) t_1))
       (/ 2.0 (* (/ (* t_1 (* t k_m)) (* l l)) k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if (k_m <= 3e-69) {
		tmp = 2.0 / ((((fma(-0.3333333333333333, (k_m * k_m), 1.0) * t) * k_m) * k_m) * ((k_m / (cos(k_m) * l)) * (k_m / l)));
	} else if (k_m <= 2.1e+115) {
		tmp = l / ((((k_m * k_m) * t) / (l * 2.0)) * t_1);
	} else {
		tmp = 2.0 / (((t_1 * (t * k_m)) / (l * l)) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (k_m <= 3e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k_m * k_m), 1.0) * t) * k_m) * k_m) * Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(k_m / l))));
	elseif (k_m <= 2.1e+115)
		tmp = Float64(l / Float64(Float64(Float64(Float64(k_m * k_m) * t) / Float64(l * 2.0)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(t * k_m)) / Float64(l * l)) * 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[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 3e-69], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.1e+115], N[(l / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$1 * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.99999999999999989e-69

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 69.4

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 75.9%

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

   if 2.99999999999999989e-69 < k < 2.10000000000000003e115

   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

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 77.7%

      \[\leadsto \frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.8%

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

   if 2.10000000000000003e115 < k

   1. Initial program 32.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 79.1%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}{\cos k \cdot \ell}}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.2%

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

   11. Applied rewrites 87.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{\ell}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot 2} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot k\right)}{\ell \cdot \ell} \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k\_m \cdot k\_m, 1\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos k\_m \cdot 2\right) \cdot \ell}{\left(\left(t \cdot k\_m\right) \cdot k\_m\right) \cdot \left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.4e-6)
   (/
    2.0
    (*
     (* (* (* (fma -0.3333333333333333 (* k_m k_m) 1.0) t) k_m) k_m)
     (* (/ k_m (* (cos k_m) l)) (/ k_m l))))
   (*
    (/
     (* (* (cos k_m) 2.0) l)
     (* (* (* t k_m) k_m) (- 0.5 (* (cos (+ k_m k_m)) 0.5))))
    l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.4e-6) {
		tmp = 2.0 / ((((fma(-0.3333333333333333, (k_m * k_m), 1.0) * t) * k_m) * k_m) * ((k_m / (cos(k_m) * l)) * (k_m / l)));
	} else {
		tmp = (((cos(k_m) * 2.0) * l) / (((t * k_m) * k_m) * (0.5 - (cos((k_m + k_m)) * 0.5)))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.4e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k_m * k_m), 1.0) * t) * k_m) * k_m) * Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(k_m / l))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * 2.0) * l) / Float64(Float64(Float64(t * k_m) * k_m) * Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.4e-6], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(t * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4000000000000002e-6

   1. Initial program 39.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 71.0

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 77.1%

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

   if 4.4000000000000002e-6 < k

   1. Initial program 24.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 85.3%

      \[\leadsto \frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k\_m \cdot k\_m, 1\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\ell \cdot 2} \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e-69)
   (/
    2.0
    (*
     (* (* (* (fma -0.3333333333333333 (* k_m k_m) 1.0) t) k_m) k_m)
     (* (/ k_m (* (cos k_m) l)) (/ k_m l))))
   (/ l (* (/ (* (* k_m k_m) t) (* l 2.0)) (* (sin k_m) (tan k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-69) {
		tmp = 2.0 / ((((fma(-0.3333333333333333, (k_m * k_m), 1.0) * t) * k_m) * k_m) * ((k_m / (cos(k_m) * l)) * (k_m / l)));
	} else {
		tmp = l / ((((k_m * k_m) * t) / (l * 2.0)) * (sin(k_m) * tan(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k_m * k_m), 1.0) * t) * k_m) * k_m) * Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(k_m / l))));
	else
		tmp = Float64(l / Float64(Float64(Float64(Float64(k_m * k_m) * t) / Float64(l * 2.0)) * Float64(sin(k_m) * tan(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e-69], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.99999999999999989e-69

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 69.4

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 75.9%

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

   if 2.99999999999999989e-69 < k

   1. Initial program 27.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 78.4%

      \[\leadsto \frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot 2} \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / ((k_m / l) * ((t * k_m) * (math.sin(k_m) * (math.tan(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(t * k_m) * Float64(sin(k_m) * Float64(tan(k_m) / l)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((k_m / l) * ((t * k_m) * (sin(k_m) * (tan(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[(t * k$95$m), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lower-sin.f64 72.2

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

5. Applied rewrites 72.2%

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

7. Applied rewrites 76.3%

   \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}{\cos k \cdot \ell}}{\color{blue}{\ell}}} \]
8. Step-by-step derivation

9. Applied rewrites 92.8%

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

11. Applied rewrites 96.1%

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

12. Final simplification 96.1%

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

Alternative 8: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k\_m \cdot k\_m, 1\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)} \cdot 2\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.6e-37)
   (/
    2.0
    (*
     (* (* (* (fma -0.3333333333333333 (* k_m k_m) 1.0) t) k_m) k_m)
     (* (/ k_m (* (cos k_m) l)) (/ k_m l))))
   (* (* (/ l (* (* (* k_m k_m) t) (* (sin k_m) (tan k_m)))) 2.0) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.6e-37) {
		tmp = 2.0 / ((((fma(-0.3333333333333333, (k_m * k_m), 1.0) * t) * k_m) * k_m) * ((k_m / (cos(k_m) * l)) * (k_m / l)));
	} else {
		tmp = ((l / (((k_m * k_m) * t) * (sin(k_m) * tan(k_m)))) * 2.0) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.6e-37)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k_m * k_m), 1.0) * t) * k_m) * k_m) * Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(k_m / l))));
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(Float64(k_m * k_m) * t) * Float64(sin(k_m) * tan(k_m)))) * 2.0) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.6e-37], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.60000000000000007e-37

   1. Initial program 40.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 76.7%

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

   if 3.60000000000000007e-37 < k

   1. Initial program 24.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 80.6%

      \[\leadsto \frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot 2\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k\_m \cdot k\_m, 1\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{elif}\;k\_m \leq 6.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.3333333333333333, k\_m \cdot k\_m, 1\right)}{k\_m \cdot k\_m} \cdot \frac{\left(\cos k\_m \cdot 2\right) \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.4e-78)
   (/
    2.0
    (*
     (* (* (* (fma -0.3333333333333333 (* k_m k_m) 1.0) t) k_m) k_m)
     (* (/ k_m (* (cos k_m) l)) (/ k_m l))))
   (if (<= k_m 6.6e+80)
     (*
      (/ (* (/ l t) (fma 0.3333333333333333 (* k_m k_m) 1.0)) (* k_m k_m))
      (/ (* (* (cos k_m) 2.0) l) (* k_m k_m)))
     (*
      (/ (* 2.0 l) (* (* (* k_m k_m) t) (- 0.5 (* (cos (+ k_m k_m)) 0.5))))
      l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.4e-78) {
		tmp = 2.0 / ((((fma(-0.3333333333333333, (k_m * k_m), 1.0) * t) * k_m) * k_m) * ((k_m / (cos(k_m) * l)) * (k_m / l)));
	} else if (k_m <= 6.6e+80) {
		tmp = (((l / t) * fma(0.3333333333333333, (k_m * k_m), 1.0)) / (k_m * k_m)) * (((cos(k_m) * 2.0) * l) / (k_m * k_m));
	} else {
		tmp = ((2.0 * l) / (((k_m * k_m) * t) * (0.5 - (cos((k_m + k_m)) * 0.5)))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.4e-78)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k_m * k_m), 1.0) * t) * k_m) * k_m) * Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(k_m / l))));
	elseif (k_m <= 6.6e+80)
		tmp = Float64(Float64(Float64(Float64(l / t) * fma(0.3333333333333333, Float64(k_m * k_m), 1.0)) / Float64(k_m * k_m)) * Float64(Float64(Float64(cos(k_m) * 2.0) * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.4e-78], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6.6e+80], N[(N[(N[(N[(l / t), $MachinePrecision] * N[(0.3333333333333333 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.39999999999999987e-78

   1. Initial program 39.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 75.6%

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

   if 5.39999999999999987e-78 < k < 6.59999999999999982e80

   1. Initial program 27.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 69.5%

      \[\leadsto \frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.8%

      \[\leadsto \frac{\left(2 \cdot \cos k\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\mathsf{fma}\left(-0.5, \cos \left(k + k\right), 0.5\right) \cdot t}} \]

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 70.8%

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

   if 6.59999999999999982e80 < k

   1. Initial program 28.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 83.0%

      \[\leadsto \frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\ell} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 68.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right)}{k \cdot k} \cdot \frac{\left(\cos k \cdot 2\right) \cdot \ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+257}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - k\_m \cdot k\_m}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1.15e+257)
   (* (/ (* l 2.0) (* (* k_m k_m) t)) (/ l (* k_m k_m)))
   (*
    (/
     (- 2.0 (* k_m k_m))
     (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))
    (* l l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.15e+257) {
		tmp = ((l * 2.0) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
	} else {
		tmp = ((2.0 - (k_m * k_m)) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)) * (l * l);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 1.15d+257) then
        tmp = ((l * 2.0d0) / ((k_m * k_m) * t)) * (l / (k_m * k_m))
    else
        tmp = ((2.0d0 - (k_m * k_m)) / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m) * k_m)) * (l * l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.15e+257) {
		tmp = ((l * 2.0) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
	} else {
		tmp = ((2.0 - (k_m * k_m)) / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)) * (l * l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1.15e+257:
		tmp = ((l * 2.0) / ((k_m * k_m) * t)) * (l / (k_m * k_m))
	else:
		tmp = ((2.0 - (k_m * k_m)) / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)) * (l * l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 1.15e+257)
		tmp = Float64(Float64(Float64(l * 2.0) / Float64(Float64(k_m * k_m) * t)) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 - Float64(k_m * k_m)) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m)) * Float64(l * l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 1.15e+257)
		tmp = ((l * 2.0) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
	else
		tmp = ((2.0 - (k_m * k_m)) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)) * (l * l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1.15e+257], N[(N[(N[(l * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 - N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e257

   1. Initial program 36.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 72.4%

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

   if 1.15e257 < l

   1. Initial program 20.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.0%

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

   10. Applied rewrites 60.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+257}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - k \cdot k}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 2.9× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 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) {
	return 2.0 / (((k_m * k_m) * t) * ((k_m / (cos(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 / (cos(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 / (Math.cos(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 / (math.cos(k_m) * l)) * (k_m / l)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(Float64(k_m / Float64(cos(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 / (cos(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[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lower-sin.f64 72.2

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

5. Applied rewrites 72.2%

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

7. Applied rewrites 91.7%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 72.1%

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

11. Final simplification 72.1%

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

Alternative 12: 73.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 71.4%

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

8. Final simplification 71.4%

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

Alternative 13: 71.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

9. Applied rewrites 69.4%

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

11. Applied rewrites 70.6%

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

12. Final simplification 70.6%

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

Alternative 14: 71.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

9. Applied rewrites 69.4%

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

11. Applied rewrites 70.5%

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

12. Final simplification 70.5%

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

Alternative 15: 69.5% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

9. Applied rewrites 69.4%

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

11. Applied rewrites 69.4%

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

12. Final simplification 69.4%

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

Alternative 16: 70.2% accurate, 11.0× speedup

Math

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

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(Float64(Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(t * k_m)) * k_m)) * 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[(N[(N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]

Derivation

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

9. Applied rewrites 69.4%

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

10. Final simplification 69.4%

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

Alternative 17: 69.5% accurate, 11.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

8. Final simplification 69.3%

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

Reproduce

herbie shell --seed 2024235 
(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))))