Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 98.4%

Time: 16.6s

Alternatives: 19

Speedup: 11.0×

• 39.7% 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: 35.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k\_m}{\ell} \cdot \left(\sin k\_m \cdot t\right)\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot \frac{k\_m}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.8e+53)
   (/ 2.0 (* (* (* (/ (tan k_m) l) (* (sin k_m) t)) k_m) (/ 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 <= 6.8e+53) {
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (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 <= 6.8d+53) then
        tmp = 2.0d0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (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 <= 6.8e+53) {
		tmp = 2.0 / ((((Math.tan(k_m) / l) * (Math.sin(k_m) * t)) * k_m) * (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 <= 6.8e+53:
		tmp = 2.0 / ((((math.tan(k_m) / l) * (math.sin(k_m) * t)) * k_m) * (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 <= 6.8e+53)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) / l) * Float64(sin(k_m) * t)) * k_m) * Float64(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 <= 6.8e+53)
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (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, 6.8e+53], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $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 < 6.79999999999999995e53

   1. Initial program 37.3%

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

   3. Taylor expanded in t around 0

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

      1. *-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 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 92.1%

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

   11. Applied rewrites 94.1%

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

   if 6.79999999999999995e53 < k

   1. Initial program 39.9%

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

   3. Taylor expanded in t around 0

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

      1. *-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 98.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \left(\sin k \cdot t\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\cos k \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k\_m}{\ell} \cdot \left(\sin k\_m \cdot t\right)\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k\_m}{\cos k\_m \cdot \ell}\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.8e+53)
   (/ 2.0 (* (* (* (/ (tan k_m) l) (* (sin k_m) t)) k_m) (/ k_m l)))
   (/
    2.0
    (*
     (* (* (- 0.5 (* (cos (+ k_m k_m)) 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 <= 6.8e+53) {
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l));
	} else {
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / (cos(k_m) * l))) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.8d+53) then
        tmp = 2.0d0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l))
    else
        tmp = 2.0d0 / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * (k_m / (cos(k_m) * l))) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.8e+53) {
		tmp = 2.0 / ((((Math.tan(k_m) / l) * (Math.sin(k_m) * t)) * k_m) * (k_m / l));
	} else {
		tmp = 2.0 / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / (Math.cos(k_m) * l))) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.8e+53:
		tmp = 2.0 / ((((math.tan(k_m) / l) * (math.sin(k_m) * t)) * k_m) * (k_m / l))
	else:
		tmp = 2.0 / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / (math.cos(k_m) * l))) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.8e+53)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) / l) * Float64(sin(k_m) * t)) * k_m) * Float64(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 / Float64(cos(k_m) * l))) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 6.8e+53)
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l));
	else
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / (cos(k_m) * l))) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.8e+53], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $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 / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.79999999999999995e53

   1. Initial program 37.3%

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

   3. Taylor expanded in t around 0

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

      1. *-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 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 92.1%

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

   11. Applied rewrites 94.1%

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

   if 6.79999999999999995e53 < k

   1. Initial program 39.9%

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

   3. Taylor expanded in t around 0

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

      1. *-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 98.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \left(\sin k \cdot t\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k\_m}{\ell} \cdot \left(\sin k\_m \cdot t\right)\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.5, \cos \left(k\_m + k\_m\right), 0.5\right) \cdot \left(\frac{k\_m}{\ell} \cdot t\right)\right) \cdot \frac{k\_m}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.8e+53)
   (/ 2.0 (* (* (* (/ (tan k_m) l) (* (sin k_m) t)) k_m) (/ 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 <= 6.8e+53) {
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (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 <= 6.8e+53)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) / l) * Float64(sin(k_m) * t)) * k_m) * Float64(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, 6.8e+53], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $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 < 6.79999999999999995e53

   1. Initial program 37.3%

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

   3. Taylor expanded in t around 0

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

      1. *-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 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 92.1%

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

   11. Applied rewrites 94.1%

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

   if 6.79999999999999995e53 < k

   1. Initial program 39.9%

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

   3. Taylor expanded in t around 0

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

      1. *-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 94.4%

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

   9. Applied rewrites 98.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \left(\sin k \cdot t\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-0.5, \cos \left(k + k\right), 0.5\right) \cdot \left(\frac{k}{\ell} \cdot t\right)\right) \cdot \frac{k}{\cos k \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot t\\ \mathbf{if}\;t \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t\_1 \cdot k\_m\right) \cdot \tan k\_m\right) \cdot \frac{1}{\ell}\right) \cdot k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k\_m}{\ell} \cdot t\_1\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) t)))
   (if (<= t 2.3e-133)
     (/ 2.0 (/ (* (* (* (* t_1 k_m) (tan k_m)) (/ 1.0 l)) k_m) l))
     (/ 2.0 (* (* (* (/ (tan k_m) l) t_1) k_m) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * t;
	double tmp;
	if (t <= 2.3e-133) {
		tmp = 2.0 / (((((t_1 * k_m) * tan(k_m)) * (1.0 / l)) * k_m) / l);
	} else {
		tmp = 2.0 / ((((tan(k_m) / l) * t_1) * k_m) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * t
    if (t <= 2.3d-133) then
        tmp = 2.0d0 / (((((t_1 * k_m) * tan(k_m)) * (1.0d0 / l)) * k_m) / l)
    else
        tmp = 2.0d0 / ((((tan(k_m) / l) * t_1) * k_m) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * t;
	double tmp;
	if (t <= 2.3e-133) {
		tmp = 2.0 / (((((t_1 * k_m) * Math.tan(k_m)) * (1.0 / l)) * k_m) / l);
	} else {
		tmp = 2.0 / ((((Math.tan(k_m) / l) * t_1) * k_m) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * t
	tmp = 0
	if t <= 2.3e-133:
		tmp = 2.0 / (((((t_1 * k_m) * math.tan(k_m)) * (1.0 / l)) * k_m) / l)
	else:
		tmp = 2.0 / ((((math.tan(k_m) / l) * t_1) * k_m) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * t)
	tmp = 0.0
	if (t <= 2.3e-133)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_1 * k_m) * tan(k_m)) * Float64(1.0 / l)) * k_m) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) / l) * t_1) * k_m) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * t;
	tmp = 0.0;
	if (t <= 2.3e-133)
		tmp = 2.0 / (((((t_1 * k_m) * tan(k_m)) * (1.0 / l)) * k_m) / l);
	else
		tmp = 2.0 / ((((tan(k_m) / l) * t_1) * k_m) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, 2.3e-133], N[(2.0 / N[(N[(N[(N[(N[(t$95$1 * k$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e-133

   1. Initial program 35.9%

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

   3. Taylor expanded in t around 0

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

      1. *-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 91.1%

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

   9. Applied rewrites 91.4%

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

   11. Applied rewrites 93.0%

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

   if 2.3e-133 < t

   1. Initial program 41.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 85.2

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 92.5%

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

   11. Applied rewrites 97.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\left(\sin k \cdot t\right) \cdot k\right) \cdot \tan k\right) \cdot \frac{1}{\ell}\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \left(\sin k \cdot t\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot t\\ \mathbf{if}\;t \leq 3 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_1 \cdot k\_m\right) \cdot \tan k\_m\right) \cdot \frac{k\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k\_m}{\ell} \cdot t\_1\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) t)))
   (if (<= t 3e-134)
     (/ 2.0 (/ (* (* (* t_1 k_m) (tan k_m)) (/ k_m l)) l))
     (/ 2.0 (* (* (* (/ (tan k_m) l) t_1) k_m) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * t;
	double tmp;
	if (t <= 3e-134) {
		tmp = 2.0 / ((((t_1 * k_m) * tan(k_m)) * (k_m / l)) / l);
	} else {
		tmp = 2.0 / ((((tan(k_m) / l) * t_1) * k_m) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * t
    if (t <= 3d-134) then
        tmp = 2.0d0 / ((((t_1 * k_m) * tan(k_m)) * (k_m / l)) / l)
    else
        tmp = 2.0d0 / ((((tan(k_m) / l) * t_1) * k_m) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * t;
	double tmp;
	if (t <= 3e-134) {
		tmp = 2.0 / ((((t_1 * k_m) * Math.tan(k_m)) * (k_m / l)) / l);
	} else {
		tmp = 2.0 / ((((Math.tan(k_m) / l) * t_1) * k_m) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * t
	tmp = 0
	if t <= 3e-134:
		tmp = 2.0 / ((((t_1 * k_m) * math.tan(k_m)) * (k_m / l)) / l)
	else:
		tmp = 2.0 / ((((math.tan(k_m) / l) * t_1) * k_m) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * t)
	tmp = 0.0
	if (t <= 3e-134)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_1 * k_m) * tan(k_m)) * Float64(k_m / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) / l) * t_1) * k_m) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * t;
	tmp = 0.0;
	if (t <= 3e-134)
		tmp = 2.0 / ((((t_1 * k_m) * tan(k_m)) * (k_m / l)) / l);
	else
		tmp = 2.0 / ((((tan(k_m) / l) * t_1) * k_m) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, 3e-134], N[(2.0 / N[(N[(N[(N[(t$95$1 * k$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3e-134

   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 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 91.1%

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

   9. Applied rewrites 91.4%

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

   11. Applied rewrites 93.0%

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

   if 3e-134 < t

   1. Initial program 40.9%

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

   3. Taylor expanded in t around 0

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

      1. *-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 85.4

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

   5. Applied rewrites 85.4%

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

   7. Applied rewrites 93.8%

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

   9. Applied rewrites 92.6%

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

   11. Applied rewrites 97.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\sin k \cdot t\right) \cdot k\right) \cdot \tan k\right) \cdot \frac{k}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \left(\sin k \cdot t\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot t\right) \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k\_m}{\ell} \cdot \left(\sin k\_m \cdot t\right)\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.3e-230)
   (/ 2.0 (/ (* (* (* (tan k_m) (sin k_m)) t) (* (/ k_m l) k_m)) l))
   (/ 2.0 (* (* (* (/ (tan k_m) l) (* (sin k_m) t)) k_m) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-230) {
		tmp = 2.0 / ((((tan(k_m) * sin(k_m)) * t) * ((k_m / l) * k_m)) / l);
	} else {
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.3d-230) then
        tmp = 2.0d0 / ((((tan(k_m) * sin(k_m)) * t) * ((k_m / l) * k_m)) / l)
    else
        tmp = 2.0d0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-230) {
		tmp = 2.0 / ((((Math.tan(k_m) * Math.sin(k_m)) * t) * ((k_m / l) * k_m)) / l);
	} else {
		tmp = 2.0 / ((((Math.tan(k_m) / l) * (Math.sin(k_m) * t)) * k_m) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.3e-230:
		tmp = 2.0 / ((((math.tan(k_m) * math.sin(k_m)) * t) * ((k_m / l) * k_m)) / l)
	else:
		tmp = 2.0 / ((((math.tan(k_m) / l) * (math.sin(k_m) * t)) * k_m) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.3e-230)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) * sin(k_m)) * t) * Float64(Float64(k_m / l) * k_m)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) / l) * Float64(sin(k_m) * t)) * k_m) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.3e-230)
		tmp = 2.0 / ((((tan(k_m) * sin(k_m)) * t) * ((k_m / l) * k_m)) / l);
	else
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.3e-230], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.2999999999999998e-230

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 92.7%

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

   if 2.2999999999999998e-230 < t

   1. Initial program 38.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 91.5%

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

   9. Applied rewrites 90.5%

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

   11. Applied rewrites 95.6%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \left(\sin k \cdot t\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m}{\ell} \cdot k\_m}{\ell} \cdot \left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k\_m}{\ell} \cdot \left(\sin k\_m \cdot t\right)\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6.6e-231)
   (/ 2.0 (* (/ (* (/ k_m l) k_m) l) (* (* (tan k_m) (sin k_m)) t)))
   (/ 2.0 (* (* (* (/ (tan k_m) l) (* (sin k_m) t)) k_m) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6.6e-231) {
		tmp = 2.0 / ((((k_m / l) * k_m) / l) * ((tan(k_m) * sin(k_m)) * t));
	} else {
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 6.6d-231) then
        tmp = 2.0d0 / ((((k_m / l) * k_m) / l) * ((tan(k_m) * sin(k_m)) * t))
    else
        tmp = 2.0d0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6.6e-231) {
		tmp = 2.0 / ((((k_m / l) * k_m) / l) * ((Math.tan(k_m) * Math.sin(k_m)) * t));
	} else {
		tmp = 2.0 / ((((Math.tan(k_m) / l) * (Math.sin(k_m) * t)) * k_m) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 6.6e-231:
		tmp = 2.0 / ((((k_m / l) * k_m) / l) * ((math.tan(k_m) * math.sin(k_m)) * t))
	else:
		tmp = 2.0 / ((((math.tan(k_m) / l) * (math.sin(k_m) * t)) * k_m) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 6.6e-231)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / l) * k_m) / l) * Float64(Float64(tan(k_m) * sin(k_m)) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) / l) * Float64(sin(k_m) * t)) * k_m) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 6.6e-231)
		tmp = 2.0 / ((((k_m / l) * k_m) / l) * ((tan(k_m) * sin(k_m)) * t));
	else
		tmp = 2.0 / ((((tan(k_m) / l) * (sin(k_m) * t)) * k_m) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6.6e-231], N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.60000000000000056e-231

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 91.2%

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

   if 6.60000000000000056e-231 < t

   1. Initial program 38.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 91.5%

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

   9. Applied rewrites 90.5%

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

   11. Applied rewrites 95.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot k}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \left(\sin k \cdot t\right)\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k\_m \cdot k\_m, 1\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right) \cdot \frac{k\_m}{\ell}\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.75e-106)
   (/
    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))))
   (/ 2.0 (* (* (* (/ t l) (* (tan k_m) (sin k_m))) (/ k_m l)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.75e-106) {
		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 = 2.0 / ((((t / l) * (tan(k_m) * sin(k_m))) * (k_m / l)) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.75e-106)
		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(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(tan(k_m) * sin(k_m))) * Float64(k_m / l)) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.75e-106], 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[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e-106

   1. Initial program 37.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 93.9%

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

   8. 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 78.1%

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

   if 1.75e-106 < k

   1. Initial program 37.8%

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

   3. Taylor expanded in t around 0

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

      1. *-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 73.8

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 87.9%

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

   9. Applied rewrites 91.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k}{\ell}\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot t, 0.16666666666666666, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell \cdot \ell} \cdot \left(\left(\left(\sin k\_m \cdot t\right) \cdot k\_m\right) \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.02e-9)
   (/
    2.0
    (/
     (*
      (* (* (fma (* (* k_m k_m) t) 0.16666666666666666 t) k_m) k_m)
      (* (/ k_m l) k_m))
     l))
   (/ 2.0 (* (/ k_m (* l l)) (* (* (* (sin k_m) t) k_m) (tan k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.02e-9) {
		tmp = 2.0 / ((((fma(((k_m * k_m) * t), 0.16666666666666666, t) * k_m) * k_m) * ((k_m / l) * k_m)) / l);
	} else {
		tmp = 2.0 / ((k_m / (l * l)) * (((sin(k_m) * t) * k_m) * tan(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.02e-9)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) * t), 0.16666666666666666, t) * k_m) * k_m) * Float64(Float64(k_m / l) * k_m)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / Float64(l * l)) * Float64(Float64(Float64(sin(k_m) * t) * 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, 1.02e-9], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * 0.16666666666666666 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.01999999999999999e-9

   1. Initial program 37.4%

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

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 91.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 79.7%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(t \cdot \left(k \cdot k\right), 0.16666666666666666, t\right) \cdot k\right) \cdot k\right)}{\ell}} \]

   if 1.01999999999999999e-9 < k

   1. Initial program 38.9%

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

   3. Taylor expanded in t around 0

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

      1. *-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 72.7

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 92.1%

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

   11. Applied rewrites 74.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.16666666666666666, t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell \cdot \ell} \cdot \left(\left(\left(\sin k \cdot t\right) \cdot k\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot t, 0.16666666666666666, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k\_m \cdot t\right) \cdot \sin k\_m}{\ell \cdot \ell} \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.02e-9)
   (/
    2.0
    (/
     (*
      (* (* (fma (* (* k_m k_m) t) 0.16666666666666666 t) k_m) k_m)
      (* (/ k_m l) k_m))
     l))
   (/ 2.0 (* (/ (* (* (tan k_m) t) (sin k_m)) (* l l)) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.02e-9) {
		tmp = 2.0 / ((((fma(((k_m * k_m) * t), 0.16666666666666666, t) * k_m) * k_m) * ((k_m / l) * k_m)) / l);
	} else {
		tmp = 2.0 / ((((tan(k_m) * t) * sin(k_m)) / (l * l)) * (k_m * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.02e-9)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) * t), 0.16666666666666666, t) * k_m) * k_m) * Float64(Float64(k_m / l) * k_m)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) * t) * sin(k_m)) / Float64(l * l)) * Float64(k_m * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.02e-9], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * 0.16666666666666666 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * t), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.01999999999999999e-9

   1. Initial program 37.4%

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

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 91.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 79.7%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(t \cdot \left(k \cdot k\right), 0.16666666666666666, t\right) \cdot k\right) \cdot k\right)}{\ell}} \]

   if 1.01999999999999999e-9 < k

   1. Initial program 38.9%

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

   3. Taylor expanded in t around 0

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

      1. *-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 72.7

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 92.1%

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

   11. Applied rewrites 74.1%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.16666666666666666, t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot t\right) \cdot \sin k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((((tan(k_m) / l) * (sin(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 = 2.0d0 / ((((tan(k_m) / l) * (sin(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 2.0 / ((((Math.tan(k_m) / l) * (Math.sin(k_m) * t)) * k_m) * (k_m / l));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 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 92.0%

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

9. Applied rewrites 91.8%

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

11. Applied rewrites 94.2%

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

12. Final simplification 94.2%

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

Alternative 12: 94.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 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 92.0%

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

9. Applied rewrites 91.8%

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

11. Applied rewrites 92.7%

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

12. Final simplification 92.7%

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

Alternative 13: 73.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \left(k\_m \cdot k\_m\right) \cdot t\\ \mathbf{if}\;\ell \leq 3.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(t\_1, 0.16666666666666666, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{k\_m \cdot k\_m}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (* k_m k_m) t)))
   (if (<= l 3.4e+189)
     (/
      2.0
      (/
       (* (* (* (fma t_1 0.16666666666666666 t) k_m) k_m) (* (/ k_m l) k_m))
       l))
     (/ 2.0 (* t_1 (/ (* k_m k_m) (* (* (cos k_m) l) l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m * k_m) * t;
	double tmp;
	if (l <= 3.4e+189) {
		tmp = 2.0 / ((((fma(t_1, 0.16666666666666666, t) * k_m) * k_m) * ((k_m / l) * k_m)) / l);
	} else {
		tmp = 2.0 / (t_1 * ((k_m * k_m) / ((cos(k_m) * l) * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m * k_m) * t)
	tmp = 0.0
	if (l <= 3.4e+189)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(t_1, 0.16666666666666666, t) * k_m) * k_m) * Float64(Float64(k_m / l) * k_m)) / l));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k_m * k_m) / Float64(Float64(cos(k_m) * l) * l))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[l, 3.4e+189], N[(2.0 / N[(N[(N[(N[(N[(t$95$1 * 0.16666666666666666 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.39999999999999983e189

   1. Initial program 37.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 91.6%

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

   9. Applied rewrites 92.6%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 77.1%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(t \cdot \left(k \cdot k\right), 0.16666666666666666, t\right) \cdot k\right) \cdot k\right)}{\ell}} \]

   if 3.39999999999999983e189 < l

   1. Initial program 45.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 68.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.16666666666666666, t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot t, 0.16666666666666666, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)}{\ell}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (/
   (*
    (* (* (fma (* (* k_m k_m) t) 0.16666666666666666 t) k_m) k_m)
    (* (/ k_m l) k_m))
   l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((((fma(((k_m * k_m) * t), 0.16666666666666666, t) * k_m) * k_m) * ((k_m / l) * k_m)) / l);
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) * t), 0.16666666666666666, t) * k_m) * k_m) * Float64(Float64(k_m / l) * k_m)) / l))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * 0.16666666666666666 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 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 92.0%

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

9. Applied rewrites 91.8%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 75.3%

    \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(t \cdot \left(k \cdot k\right), 0.16666666666666666, t\right) \cdot k\right) \cdot k\right)}{\ell}} \]

13. Final simplification 75.3%

    \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.16666666666666666, t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}{\ell}} \]
14. Add Preprocessing

Alternative 15: 72.9% 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 37.7%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
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 66.3

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

5. Applied rewrites 66.3%

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

7. Applied rewrites 75.3%

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

8. Final simplification 75.3%

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

Alternative 16: 69.7% 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 37.7%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
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 66.3

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

5. Applied rewrites 66.3%

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

7. Applied rewrites 72.5%

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

8. Final simplification 72.5%

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

Alternative 17: 20.8% accurate, 16.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.11666666666666667}{\frac{t}{\ell \cdot \ell}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (/ -0.11666666666666667 (/ t (* l l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.11666666666666667 / (t / (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 = (-0.11666666666666667d0) / (t / (l * l))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.11666666666666667 / (t / (l * l));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.11666666666666667 / (t / (l * l))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.11666666666666667 / Float64(t / Float64(l * l)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.11666666666666667 / (t / (l * l));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.11666666666666667 / N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

5. Applied rewrites 30.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.11666666666666667, \frac{\ell \cdot \ell}{t}, \left(\left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right) \cdot -2\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(-0.3333333333333333, k \cdot k, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-7}{60} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 14.5%

   \[\leadsto \left(\frac{\ell \cdot \ell}{t} \cdot \left(\frac{-0.11666666666666667}{k \cdot k} + -0.0205026455026455\right)\right) \cdot \color{blue}{\left(k \cdot k\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]
10. Step-by-step derivation

11. Applied rewrites 21.6%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \]
12. Step-by-step derivation

13. Applied rewrites 21.7%

    \[\leadsto \frac{-0.11666666666666667}{\frac{t}{\ell \cdot \color{blue}{\ell}}} \]
14. Add Preprocessing

Alternative 18: 20.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (/ -0.11666666666666667 t) (* l l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (-0.11666666666666667 / t) * (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 = ((-0.11666666666666667d0) / t) * (l * l)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (-0.11666666666666667 / t) * (l * l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (-0.11666666666666667 / t) * (l * l)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(-0.11666666666666667 / t) * Float64(l * l))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (-0.11666666666666667 / t) * (l * l);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(-0.11666666666666667 / t), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

5. Applied rewrites 30.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.11666666666666667, \frac{\ell \cdot \ell}{t}, \left(\left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right) \cdot -2\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(-0.3333333333333333, k \cdot k, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-7}{60} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 14.5%

   \[\leadsto \left(\frac{\ell \cdot \ell}{t} \cdot \left(\frac{-0.11666666666666667}{k \cdot k} + -0.0205026455026455\right)\right) \cdot \color{blue}{\left(k \cdot k\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]
10. Step-by-step derivation

11. Applied rewrites 21.6%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \]
12. Step-by-step derivation

13. Applied rewrites 21.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]

14. Final simplification 21.6%

    \[\leadsto \frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right) \]
15. Add Preprocessing

Alternative 19: 18.1% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \cdot \ell \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (* (/ l t) -0.11666666666666667) l))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l / t) * -0.11666666666666667) * 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 / t) * (-0.11666666666666667d0)) * l
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l / t) * -0.11666666666666667) * l;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l / t) * -0.11666666666666667) * l

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l / t) * -0.11666666666666667) * l)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l / t) * -0.11666666666666667) * l;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l / t), $MachinePrecision] * -0.11666666666666667), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 37.7%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

5. Applied rewrites 30.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.11666666666666667, \frac{\ell \cdot \ell}{t}, \left(\left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right) \cdot -2\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(-0.3333333333333333, k \cdot k, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-7}{60} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 14.5%

   \[\leadsto \left(\frac{\ell \cdot \ell}{t} \cdot \left(\frac{-0.11666666666666667}{k \cdot k} + -0.0205026455026455\right)\right) \cdot \color{blue}{\left(k \cdot k\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]
10. Step-by-step derivation

11. Applied rewrites 21.6%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \]
12. Step-by-step derivation

13. Applied rewrites 19.4%

    \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \]

14. Final simplification 19.4%

    \[\leadsto \left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \cdot \ell \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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))))