Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 93.6%

Time: 17.0s

Alternatives: 15

Speedup: 11.0×

• 39.9% 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: 36.2% 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: 93.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (/ 2.0 (* (* k k) (/ (* t (/ (* (tan k) (sin k)) l)) l)))
   (/ (/ (* l 2.0) k) (* (/ (* t (sin k)) l) (* k (tan k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / ((k * k) * ((t * ((tan(k) * sin(k)) / l)) / l));
	} else {
		tmp = ((l * 2.0) / k) / (((t * sin(k)) / l) * (k * tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k * k) * ((t * ((tan(k) * sin(k)) / l)) / l))
    else
        tmp = ((l * 2.0d0) / k) / (((t * sin(k)) / l) * (k * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / ((k * k) * ((t * ((Math.tan(k) * Math.sin(k)) / l)) / l));
	} else {
		tmp = ((l * 2.0) / k) / (((t * Math.sin(k)) / l) * (k * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / ((k * k) * ((t * ((math.tan(k) * math.sin(k)) / l)) / l))
	else:
		tmp = ((l * 2.0) / k) / (((t * math.sin(k)) / l) * (k * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(Float64(tan(k) * sin(k)) / l)) / l)));
	else
		tmp = Float64(Float64(Float64(l * 2.0) / k) / Float64(Float64(Float64(t * sin(k)) / l) * Float64(k * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k * k) * ((t * ((tan(k) * sin(k)) / l)) / l));
	else
		tmp = ((l * 2.0) / k) / (((t * sin(k)) / l) * (k * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision] / N[(N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 15.1%

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

   6. Applied rewrites 86.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.5

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

   8. Applied rewrites 94.5%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 39.9%

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

   6. Applied rewrites 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 91.1

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

   8. Applied rewrites 91.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 97.0

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 96.5

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

   10. Applied rewrites 96.5%

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

4. Final simplification 96.1%

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

Alternative 2: 86.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sin k\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+267}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot t\_1}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (sin k))))
   (if (<= (* l l) 0.0)
     (/ 2.0 (* k (* k (* (/ t l) (/ (* k k) l)))))
     (if (<= (* l l) 1e+267)
       (/ 2.0 (* k (* k (/ (* (tan k) t_1) (* l l)))))
       (* (/ l (tan k)) (* l (/ 2.0 (* k (* k t_1)))))))))

C

double code(double t, double l, double k) {
	double t_1 = t * sin(k);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	} else if ((l * l) <= 1e+267) {
		tmp = 2.0 / (k * (k * ((tan(k) * t_1) / (l * l))));
	} else {
		tmp = (l / tan(k)) * (l * (2.0 / (k * (k * t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * sin(k)
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / (k * (k * ((t / l) * ((k * k) / l))))
    else if ((l * l) <= 1d+267) then
        tmp = 2.0d0 / (k * (k * ((tan(k) * t_1) / (l * l))))
    else
        tmp = (l / tan(k)) * (l * (2.0d0 / (k * (k * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t * Math.sin(k);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	} else if ((l * l) <= 1e+267) {
		tmp = 2.0 / (k * (k * ((Math.tan(k) * t_1) / (l * l))));
	} else {
		tmp = (l / Math.tan(k)) * (l * (2.0 / (k * (k * t_1))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t * math.sin(k)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))))
	elif (l * l) <= 1e+267:
		tmp = 2.0 / (k * (k * ((math.tan(k) * t_1) / (l * l))))
	else:
		tmp = (l / math.tan(k)) * (l * (2.0 / (k * (k * t_1))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t * sin(k))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t / l) * Float64(Float64(k * k) / l)))));
	elseif (Float64(l * l) <= 1e+267)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(tan(k) * t_1) / Float64(l * l)))));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(l * Float64(2.0 / Float64(k * Float64(k * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t * sin(k);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	elseif ((l * l) <= 1e+267)
		tmp = 2.0 / (k * (k * ((tan(k) * t_1) / (l * l))));
	else
		tmp = (l / tan(k)) * (l * (2.0 / (k * (k * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[(k * N[(k * N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+267], N[(2.0 / N[(k * N[(k * N[(N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(k * N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 41.3%

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

   11. Applied rewrites 87.3%

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

   if 0.0 < (*.f64 l l) < 9.9999999999999997e266

   1. Initial program 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 41.5%

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

   6. Applied rewrites 90.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 97.4%

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

   if 9.9999999999999997e266 < (*.f64 l l)

   1. Initial program 40.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 37.5%

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

   6. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 86.2

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

   8. Applied rewrites 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. *-commutative N/A

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

   10. Applied rewrites 76.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+267}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sin k\\ \mathbf{if}\;k \leq 1.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \frac{t\_1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot t\_1}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (sin k))))
   (if (<= k 1.2e-44)
     (/ 2.0 (* k (* (* k (/ k l)) (/ (* k t) l))))
     (if (<= k 1.4e+154)
       (/ (* l 2.0) (* (tan k) (* (* k k) (/ t_1 l))))
       (/ 2.0 (* k (* k (/ (* (tan k) t_1) (* l l)))))))))

C

double code(double t, double l, double k) {
	double t_1 = t * sin(k);
	double tmp;
	if (k <= 1.2e-44) {
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)));
	} else if (k <= 1.4e+154) {
		tmp = (l * 2.0) / (tan(k) * ((k * k) * (t_1 / l)));
	} else {
		tmp = 2.0 / (k * (k * ((tan(k) * t_1) / (l * l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * sin(k)
    if (k <= 1.2d-44) then
        tmp = 2.0d0 / (k * ((k * (k / l)) * ((k * t) / l)))
    else if (k <= 1.4d+154) then
        tmp = (l * 2.0d0) / (tan(k) * ((k * k) * (t_1 / l)))
    else
        tmp = 2.0d0 / (k * (k * ((tan(k) * t_1) / (l * l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t * Math.sin(k);
	double tmp;
	if (k <= 1.2e-44) {
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)));
	} else if (k <= 1.4e+154) {
		tmp = (l * 2.0) / (Math.tan(k) * ((k * k) * (t_1 / l)));
	} else {
		tmp = 2.0 / (k * (k * ((Math.tan(k) * t_1) / (l * l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t * math.sin(k)
	tmp = 0
	if k <= 1.2e-44:
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)))
	elif k <= 1.4e+154:
		tmp = (l * 2.0) / (math.tan(k) * ((k * k) * (t_1 / l)))
	else:
		tmp = 2.0 / (k * (k * ((math.tan(k) * t_1) / (l * l))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t * sin(k))
	tmp = 0.0
	if (k <= 1.2e-44)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(k * Float64(k / l)) * Float64(Float64(k * t) / l))));
	elseif (k <= 1.4e+154)
		tmp = Float64(Float64(l * 2.0) / Float64(tan(k) * Float64(Float64(k * k) * Float64(t_1 / l))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(tan(k) * t_1) / Float64(l * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t * sin(k);
	tmp = 0.0;
	if (k <= 1.2e-44)
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)));
	elseif (k <= 1.4e+154)
		tmp = (l * 2.0) / (tan(k) * ((k * k) * (t_1 / l)));
	else
		tmp = 2.0 / (k * (k * ((tan(k) * t_1) / (l * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.2e-44], N[(2.0 / N[(k * N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+154], N[(N[(l * 2.0), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.20000000000000004e-44

   1. Initial program 38.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 70.3%

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

   11. Applied rewrites 83.9%

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

   if 1.20000000000000004e-44 < k < 1.4e154

   1. Initial program 20.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 27.8%

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

   6. Applied rewrites 97.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 1.4e154 < k

   1. Initial program 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 18.5%

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

   6. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 57.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \frac{t \cdot \sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left(\tan k \cdot \sin k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-44)
   (/ 2.0 (* k (* (* k (/ k l)) (/ (* k t) l))))
   (if (<= k 1.4e+154)
     (/ (* l 2.0) (* (* (tan k) (sin k)) (* (* k k) (/ t l))))
     (/ 2.0 (* k (* k (/ (* (tan k) (* t (sin k))) (* l l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-44) {
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)));
	} else if (k <= 1.4e+154) {
		tmp = (l * 2.0) / ((tan(k) * sin(k)) * ((k * k) * (t / l)));
	} else {
		tmp = 2.0 / (k * (k * ((tan(k) * (t * sin(k))) / (l * l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-44) then
        tmp = 2.0d0 / (k * ((k * (k / l)) * ((k * t) / l)))
    else if (k <= 1.4d+154) then
        tmp = (l * 2.0d0) / ((tan(k) * sin(k)) * ((k * k) * (t / l)))
    else
        tmp = 2.0d0 / (k * (k * ((tan(k) * (t * sin(k))) / (l * l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-44) {
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)));
	} else if (k <= 1.4e+154) {
		tmp = (l * 2.0) / ((Math.tan(k) * Math.sin(k)) * ((k * k) * (t / l)));
	} else {
		tmp = 2.0 / (k * (k * ((Math.tan(k) * (t * Math.sin(k))) / (l * l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5e-44:
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)))
	elif k <= 1.4e+154:
		tmp = (l * 2.0) / ((math.tan(k) * math.sin(k)) * ((k * k) * (t / l)))
	else:
		tmp = 2.0 / (k * (k * ((math.tan(k) * (t * math.sin(k))) / (l * l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-44)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(k * Float64(k / l)) * Float64(Float64(k * t) / l))));
	elseif (k <= 1.4e+154)
		tmp = Float64(Float64(l * 2.0) / Float64(Float64(tan(k) * sin(k)) * Float64(Float64(k * k) * Float64(t / l))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(tan(k) * Float64(t * sin(k))) / Float64(l * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-44)
		tmp = 2.0 / (k * ((k * (k / l)) * ((k * t) / l)));
	elseif (k <= 1.4e+154)
		tmp = (l * 2.0) / ((tan(k) * sin(k)) * ((k * k) * (t / l)));
	else
		tmp = 2.0 / (k * (k * ((tan(k) * (t * sin(k))) / (l * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5e-44], N[(2.0 / N[(k * N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+154], N[(N[(l * 2.0), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(N[Tan[k], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.00000000000000039e-44

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 65.5

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 70.0%

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

   11. Applied rewrites 83.5%

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

   if 5.00000000000000039e-44 < k < 1.4e154

   1. Initial program 20.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 28.3%

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

   6. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. inv-pow N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 97.5%

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

   if 1.4e154 < k

   1. Initial program 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 18.5%

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

   6. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 57.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left(\tan k \cdot \sin k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\frac{k}{\ell} \cdot \left(k \cdot t\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k \cdot \left(k \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.75e-105)
   (/ 2.0 (* k (* k (/ (* (/ k l) (* k t)) l))))
   (/ (* l 2.0) (* k (* k (* (tan k) (* t (/ (sin k) l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e-105) {
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)));
	} else {
		tmp = (l * 2.0) / (k * (k * (tan(k) * (t * (sin(k) / l)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d-105) then
        tmp = 2.0d0 / (k * (k * (((k / l) * (k * t)) / l)))
    else
        tmp = (l * 2.0d0) / (k * (k * (tan(k) * (t * (sin(k) / l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e-105) {
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)));
	} else {
		tmp = (l * 2.0) / (k * (k * (Math.tan(k) * (t * (Math.sin(k) / l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.75e-105:
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)))
	else:
		tmp = (l * 2.0) / (k * (k * (math.tan(k) * (t * (math.sin(k) / l)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.75e-105)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(Float64(k / l) * Float64(k * t)) / l))));
	else
		tmp = Float64(Float64(l * 2.0) / Float64(k * Float64(k * Float64(tan(k) * Float64(t * Float64(sin(k) / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.75e-105)
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)));
	else
		tmp = (l * 2.0) / (k * (k * (tan(k) * (t * (sin(k) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.75e-105], N[(2.0 / N[(k * N[(k * N[(N[(N[(k / l), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e-105

   1. Initial program 38.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 66.1

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

   7. Applied rewrites 66.1%

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

   9. Applied rewrites 70.3%

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

   11. Applied rewrites 84.3%

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

   if 1.75e-105 < k

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 27.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.1

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

   10. Applied rewrites 88.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\frac{k}{\ell} \cdot \left(k \cdot t\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k \cdot \left(k \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\frac{k}{\ell} \cdot \left(k \cdot t\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.5e-104)
   (/ 2.0 (* k (* k (/ (* (/ k l) (* k t)) l))))
   (* (/ l (tan k)) (* l (/ 2.0 (* k (* k (* t (sin k)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.5e-104) {
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)));
	} else {
		tmp = (l / tan(k)) * (l * (2.0 / (k * (k * (t * sin(k))))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.5d-104) then
        tmp = 2.0d0 / (k * (k * (((k / l) * (k * t)) / l)))
    else
        tmp = (l / tan(k)) * (l * (2.0d0 / (k * (k * (t * sin(k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.5e-104) {
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)));
	} else {
		tmp = (l / Math.tan(k)) * (l * (2.0 / (k * (k * (t * Math.sin(k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.5e-104:
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)))
	else:
		tmp = (l / math.tan(k)) * (l * (2.0 / (k * (k * (t * math.sin(k))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.5e-104)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(Float64(k / l) * Float64(k * t)) / l))));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(l * Float64(2.0 / Float64(k * Float64(k * Float64(t * sin(k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.5e-104)
		tmp = 2.0 / (k * (k * (((k / l) * (k * t)) / l)));
	else
		tmp = (l / tan(k)) * (l * (2.0 / (k * (k * (t * sin(k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.5e-104], N[(2.0 / N[(k * N[(k * N[(N[(N[(k / l), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.50000000000000029e-104

   1. Initial program 38.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 66.1

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

   7. Applied rewrites 66.1%

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

   9. Applied rewrites 70.3%

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

   11. Applied rewrites 84.3%

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

   if 3.50000000000000029e-104 < k

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 27.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. *-commutative N/A

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

   10. Applied rewrites 77.9%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\frac{k}{\ell} \cdot \left(k \cdot t\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ (* l 2.0) k) (* (/ (* t (sin k)) l) (* k (tan k)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow3 N/A

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

   9. associate-/l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

4. Applied rewrites 34.6%

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

6. Applied rewrites 85.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 89.0

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

8. Applied rewrites 89.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 95.2

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 94.5

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

10. Applied rewrites 94.5%

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

11. Final simplification 94.5%

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

Alternative 8: 92.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l (* (/ (* t (sin k)) l) (* k (tan k)))) (/ 2.0 k)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow3 N/A

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

   9. associate-/l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

4. Applied rewrites 34.6%

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

6. Applied rewrites 85.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 89.0

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

8. Applied rewrites 89.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 93.0

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

10. Applied rewrites 93.0%

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

11. Final simplification 93.0%

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

Alternative 9: 73.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{\tan k \cdot \left(k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.1)
   (/ 2.0 (* k (* k (* (/ t l) (/ (* k k) l)))))
   (/ (* l 2.0) (* (tan k) (* k (/ (* k (* k t)) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.1) {
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	} else {
		tmp = (l * 2.0) / (tan(k) * (k * ((k * (k * t)) / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.1d0) then
        tmp = 2.0d0 / (k * (k * ((t / l) * ((k * k) / l))))
    else
        tmp = (l * 2.0d0) / (tan(k) * (k * ((k * (k * t)) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.1) {
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	} else {
		tmp = (l * 2.0) / (Math.tan(k) * (k * ((k * (k * t)) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.1:
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))))
	else:
		tmp = (l * 2.0) / (math.tan(k) * (k * ((k * (k * t)) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.1)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t / l) * Float64(Float64(k * k) / l)))));
	else
		tmp = Float64(Float64(l * 2.0) / Float64(tan(k) * Float64(k * Float64(Float64(k * Float64(k * t)) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.1)
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	else
		tmp = (l * 2.0) / (tan(k) * (k * ((k * (k * t)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.1], N[(2.0 / N[(k * N[(k * N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.10000000000000009

   1. Initial program 34.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 59.2

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

   7. Applied rewrites 59.2%

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

   9. Applied rewrites 62.4%

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

   11. Applied rewrites 76.3%

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

   if 2.10000000000000009 < t

   1. Initial program 33.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow3 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 36.6%

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

   6. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\frac{{k}^{3} \cdot t}{\ell}} \cdot \tan k} \]
   10. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r/ N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 76.7

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

   11. Applied rewrites 76.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{\tan k \cdot \left(k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k} \cdot \frac{\ell}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.15e-100)
   (/ 2.0 (* k (* k (* (/ t l) (/ (* k k) l)))))
   (* (/ (* l 2.0) k) (/ l (* k (* k (* k t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.15e-100) {
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	} else {
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.15d-100) then
        tmp = 2.0d0 / (k * (k * ((t / l) * ((k * k) / l))))
    else
        tmp = ((l * 2.0d0) / k) * (l / (k * (k * (k * t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.15e-100) {
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	} else {
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.15e-100:
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))))
	else:
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.15e-100)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t / l) * Float64(Float64(k * k) / l)))));
	else
		tmp = Float64(Float64(Float64(l * 2.0) / k) * Float64(l / Float64(k * Float64(k * Float64(k * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.15e-100)
		tmp = 2.0 / (k * (k * ((t / l) * ((k * k) / l))));
	else
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.15e-100], N[(2.0 / N[(k * N[(k * N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.14999999999999999e-100

   1. Initial program 30.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 59.6

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

   7. Applied rewrites 59.6%

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

   9. Applied rewrites 63.3%

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

   11. Applied rewrites 78.0%

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

   if 2.14999999999999999e-100 < t

   1. Initial program 39.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   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. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 70.4%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 74.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k} \cdot \frac{\ell}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-100}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k} \cdot \frac{\ell}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.7e-100)
   (* (* l 2.0) (/ (/ (/ l t) (* k k)) (* k k)))
   (* (/ (* l 2.0) k) (/ l (* k (* k (* k t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.7e-100) {
		tmp = (l * 2.0) * (((l / t) / (k * k)) / (k * k));
	} else {
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.7d-100) then
        tmp = (l * 2.0d0) * (((l / t) / (k * k)) / (k * k))
    else
        tmp = ((l * 2.0d0) / k) * (l / (k * (k * (k * t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.7e-100) {
		tmp = (l * 2.0) * (((l / t) / (k * k)) / (k * k));
	} else {
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.7e-100:
		tmp = (l * 2.0) * (((l / t) / (k * k)) / (k * k))
	else:
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.7e-100)
		tmp = Float64(Float64(l * 2.0) * Float64(Float64(Float64(l / t) / Float64(k * k)) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l * 2.0) / k) * Float64(l / Float64(k * Float64(k * Float64(k * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.7e-100)
		tmp = (l * 2.0) * (((l / t) / (k * k)) / (k * k));
	else
		tmp = ((l * 2.0) / k) * (l / (k * (k * (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.7e-100], N[(N[(l * 2.0), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.69999999999999988e-100

   1. Initial program 30.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   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. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 72.9%

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

   if 1.69999999999999988e-100 < t

   1. Initial program 39.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   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. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 70.4%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 74.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-100}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k} \cdot \frac{\ell}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.7% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* l 2.0) k) (/ l (* k (* k (* k t))))))

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	return ((l * 2.0) / k) * (l / (k * (k * (k * t))));
}

Python

def code(t, l, k):
	return ((l * 2.0) / k) * (l / (k * (k * (k * t))))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
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. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.1

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

5. Applied rewrites 62.1%

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

7. Applied rewrites 70.5%

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

9. Applied rewrites 71.0%

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

11. Applied rewrites 72.7%

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

12. Final simplification 72.7%

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

Alternative 13: 70.8% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l 2.0) (/ l (* (* k k) (* k (* k t))))))

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	return (l * 2.0) * (l / ((k * k) * (k * (k * t))));
}

Python

def code(t, l, k):
	return (l * 2.0) * (l / ((k * k) * (k * (k * t))))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
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. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.1

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

5. Applied rewrites 62.1%

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

7. Applied rewrites 70.5%

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

9. Applied rewrites 71.6%

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

10. Final simplification 71.6%

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

Alternative 14: 70.0% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l 2.0) (/ l (* (* k t) (* k (* k k))))))

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	return (l * 2.0) * (l / ((k * t) * (k * (k * k))));
}

Python

def code(t, l, k):
	return (l * 2.0) * (l / ((k * t) * (k * (k * k))))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(N[(k * t), $MachinePrecision] * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
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. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.1

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

5. Applied rewrites 62.1%

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

7. Applied rewrites 70.5%

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

9. Applied rewrites 71.0%

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

10. Final simplification 71.0%

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

Alternative 15: 68.8% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l 2.0) (/ l (* t (* (* k k) (* k k))))))

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	return (l * 2.0) * (l / (t * ((k * k) * (k * k))));
}

Python

def code(t, l, k):
	return (l * 2.0) * (l / (t * ((k * k) * (k * k))))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(t * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
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. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.1

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

5. Applied rewrites 62.1%

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

7. Applied rewrites 70.5%

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

8. Final simplification 70.5%

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

Reproduce

herbie shell --seed 2024232 
(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))))