Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 95.2%

Time: 15.2s

Alternatives: 13

Speedup: 8.6×

• 39.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.4% 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: 95.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2} \cdot t\\ t_2 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-268}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot t\_2\right) \cdot t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+265}:\\ \;\;\;\;\frac{2}{\frac{t\_1 \cdot k}{\ell} \cdot \frac{k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{\frac{k}{\cos k}}{\ell}\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (pow (sin k) 2.0) t)) (t_2 (* (/ k l) k)))
   (if (<= (* l l) 5e-268)
     (/ 2.0 (* (* t_2 t_2) t))
     (if (<= (* l l) 1e+265)
       (/ 2.0 (* (/ (* t_1 k) l) (/ k (* (cos k) l))))
       (/ 2.0 (* (* (/ k l) (/ (/ k (cos k)) l)) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0) * t;
	double t_2 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-268) {
		tmp = 2.0 / ((t_2 * t_2) * t);
	} else if ((l * l) <= 1e+265) {
		tmp = 2.0 / (((t_1 * k) / l) * (k / (cos(k) * l)));
	} else {
		tmp = 2.0 / (((k / l) * ((k / cos(k)) / l)) * 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) :: t_2
    real(8) :: tmp
    t_1 = (sin(k) ** 2.0d0) * t
    t_2 = (k / l) * k
    if ((l * l) <= 5d-268) then
        tmp = 2.0d0 / ((t_2 * t_2) * t)
    else if ((l * l) <= 1d+265) then
        tmp = 2.0d0 / (((t_1 * k) / l) * (k / (cos(k) * l)))
    else
        tmp = 2.0d0 / (((k / l) * ((k / cos(k)) / l)) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0) * t;
	double t_2 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-268) {
		tmp = 2.0 / ((t_2 * t_2) * t);
	} else if ((l * l) <= 1e+265) {
		tmp = 2.0 / (((t_1 * k) / l) * (k / (Math.cos(k) * l)));
	} else {
		tmp = 2.0 / (((k / l) * ((k / Math.cos(k)) / l)) * t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0) * t
	t_2 = (k / l) * k
	tmp = 0
	if (l * l) <= 5e-268:
		tmp = 2.0 / ((t_2 * t_2) * t)
	elif (l * l) <= 1e+265:
		tmp = 2.0 / (((t_1 * k) / l) * (k / (math.cos(k) * l)))
	else:
		tmp = 2.0 / (((k / l) * ((k / math.cos(k)) / l)) * t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64((sin(k) ^ 2.0) * t)
	t_2 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-268)
		tmp = Float64(2.0 / Float64(Float64(t_2 * t_2) * t));
	elseif (Float64(l * l) <= 1e+265)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * k) / l) * Float64(k / Float64(cos(k) * l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k / cos(k)) / l)) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (sin(k) ^ 2.0) * t;
	t_2 = (k / l) * k;
	tmp = 0.0;
	if ((l * l) <= 5e-268)
		tmp = 2.0 / ((t_2 * t_2) * t);
	elseif ((l * l) <= 1e+265)
		tmp = 2.0 / (((t_1 * k) / l) * (k / (cos(k) * l)));
	else
		tmp = 2.0 / (((k / l) * ((k / cos(k)) / l)) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-268], N[(2.0 / N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+265], N[(2.0 / N[(N[(N[(t$95$1 * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4.9999999999999999e-268

   1. Initial program 30.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 92.4%

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

   if 4.9999999999999999e-268 < (*.f64 l l) < 1.00000000000000007e265

   1. Initial program 30.3%

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

      1. lift-*.f64 N/A

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

      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

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

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

      11. +-rgt-identity N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. associate-*l/ N/A

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 97.7

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

   7. Applied rewrites 97.7%

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

   if 1.00000000000000007e265 < (*.f64 l l)

   1. Initial program 30.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 90.3%

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

   7. Applied rewrites 99.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-268}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+265}:\\ \;\;\;\;\frac{2}{\frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell} \cdot \frac{k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{\frac{k}{\cos k}}{\ell}\right) \cdot \left({\sin k}^{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{k}{\cos k}}{\ell}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot t\_2\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_2 \cdot t}{\ell} \cdot k\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ k (cos k)) l)) (t_2 (pow (sin k) 2.0)))
   (if (<= t 1e+54)
     (/ 2.0 (* (* (/ (* k t) l) t_2) t_1))
     (/ 2.0 (* (* (/ (* t_2 t) l) k) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = (k / cos(k)) / l;
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t <= 1e+54) {
		tmp = 2.0 / ((((k * t) / l) * t_2) * t_1);
	} else {
		tmp = 2.0 / ((((t_2 * t) / l) * 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) :: t_2
    real(8) :: tmp
    t_1 = (k / cos(k)) / l
    t_2 = sin(k) ** 2.0d0
    if (t <= 1d+54) then
        tmp = 2.0d0 / ((((k * t) / l) * t_2) * t_1)
    else
        tmp = 2.0d0 / ((((t_2 * t) / l) * k) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / Math.cos(k)) / l;
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t <= 1e+54) {
		tmp = 2.0 / ((((k * t) / l) * t_2) * t_1);
	} else {
		tmp = 2.0 / ((((t_2 * t) / l) * k) * t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / math.cos(k)) / l
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t <= 1e+54:
		tmp = 2.0 / ((((k * t) / l) * t_2) * t_1)
	else:
		tmp = 2.0 / ((((t_2 * t) / l) * k) * t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / cos(k)) / l)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t <= 1e+54)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) / l) * t_2) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_2 * t) / l) * k) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / cos(k)) / l;
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t <= 1e+54)
		tmp = 2.0 / ((((k * t) / l) * t_2) * t_1);
	else
		tmp = 2.0 / ((((t_2 * t) / l) * k) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 1e+54], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$2 * t), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.0000000000000001e54

   1. Initial program 33.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 95.4%

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

   if 1.0000000000000001e54 < t

   1. Initial program 16.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 87.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\ell} \cdot k\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3e-112)
   (/ 2.0 (* (* (* (pow (/ k l) 2.0) t) k) k))
   (/ 2.0 (* (* (* (* (tan k) (sin k)) (/ t l)) k) (/ k l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-112) {
		tmp = 2.0 / (((pow((k / l), 2.0) * t) * k) * k);
	} else {
		tmp = 2.0 / ((((tan(k) * sin(k)) * (t / l)) * k) * (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 <= 3d-112) then
        tmp = 2.0d0 / (((((k / l) ** 2.0d0) * t) * k) * k)
    else
        tmp = 2.0d0 / ((((tan(k) * sin(k)) * (t / l)) * k) * (k / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-112) {
		tmp = 2.0 / (((Math.pow((k / l), 2.0) * t) * k) * k);
	} else {
		tmp = 2.0 / ((((Math.tan(k) * Math.sin(k)) * (t / l)) * k) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3e-112:
		tmp = 2.0 / (((math.pow((k / l), 2.0) * t) * k) * k)
	else:
		tmp = 2.0 / ((((math.tan(k) * math.sin(k)) * (t / l)) * k) * (k / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3e-112)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / l) ^ 2.0) * t) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * Float64(t / l)) * k) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3e-112)
		tmp = 2.0 / (((((k / l) ^ 2.0) * t) * k) * k);
	else
		tmp = 2.0 / ((((tan(k) * sin(k)) * (t / l)) * k) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3e-112], N[(2.0 / N[(N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0000000000000001e-112

   1. Initial program 34.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 78.2%

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

   if 3.0000000000000001e-112 < k

   1. Initial program 23.6%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.0%

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

   7. Applied rewrites 86.5%

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

   9. Applied rewrites 91.8%

      \[\leadsto \frac{2}{\frac{k \cdot 1}{\frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \cdot \color{blue}{\frac{\ell}{k}}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{k}{\ell}\right)}^{-2}} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.1e-163)
   (/ 2.0 (* (/ k (pow (/ k l) -2.0)) (* k t)))
   (/ 2.0 (* (* (* (* (tan k) (sin k)) (/ t l)) (/ k l)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.1e-163) {
		tmp = 2.0 / ((k / pow((k / l), -2.0)) * (k * t));
	} else {
		tmp = 2.0 / ((((tan(k) * sin(k)) * (t / l)) * (k / l)) * 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 <= 5.1d-163) then
        tmp = 2.0d0 / ((k / ((k / l) ** (-2.0d0))) * (k * t))
    else
        tmp = 2.0d0 / ((((tan(k) * sin(k)) * (t / l)) * (k / l)) * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.1e-163) {
		tmp = 2.0 / ((k / Math.pow((k / l), -2.0)) * (k * t));
	} else {
		tmp = 2.0 / ((((Math.tan(k) * Math.sin(k)) * (t / l)) * (k / l)) * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.1e-163:
		tmp = 2.0 / ((k / math.pow((k / l), -2.0)) * (k * t))
	else:
		tmp = 2.0 / ((((math.tan(k) * math.sin(k)) * (t / l)) * (k / l)) * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.1e-163)
		tmp = Float64(2.0 / Float64(Float64(k / (Float64(k / l) ^ -2.0)) * Float64(k * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * Float64(t / l)) * Float64(k / l)) * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.1e-163)
		tmp = 2.0 / ((k / ((k / l) ^ -2.0)) * (k * t));
	else
		tmp = 2.0 / ((((tan(k) * sin(k)) * (t / l)) * (k / l)) * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.1e-163], N[(2.0 / N[(N[(k / N[Power[N[(k / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0999999999999999e-163

   1. Initial program 33.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 76.8%

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

   if 5.0999999999999999e-163 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 85.7%

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

   9. Applied rewrites 90.5%

      \[\leadsto \frac{2}{\frac{k \cdot 1}{\frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \cdot \color{blue}{\frac{\ell}{k}}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{k}{\ell}\right)}^{-2}} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k \cdot t}{\ell} \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-114)
   (/ 2.0 (* (* (* (pow (/ k l) 2.0) t) k) k))
   (/ 2.0 (* (/ k l) (* (/ (* k t) l) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-114) {
		tmp = 2.0 / (((pow((k / l), 2.0) * t) * k) * k);
	} else {
		tmp = 2.0 / ((k / l) * (((k * t) / l) * pow(sin(k), 2.0)));
	}
	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 <= 6.2d-114) then
        tmp = 2.0d0 / (((((k / l) ** 2.0d0) * t) * k) * k)
    else
        tmp = 2.0d0 / ((k / l) * (((k * t) / l) * (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-114) {
		tmp = 2.0 / (((Math.pow((k / l), 2.0) * t) * k) * k);
	} else {
		tmp = 2.0 / ((k / l) * (((k * t) / l) * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.2e-114:
		tmp = 2.0 / (((math.pow((k / l), 2.0) * t) * k) * k)
	else:
		tmp = 2.0 / ((k / l) * (((k * t) / l) * math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-114)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / l) ^ 2.0) * t) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(Float64(k * t) / l) * (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.2e-114)
		tmp = 2.0 / (((((k / l) ^ 2.0) * t) * k) * k);
	else
		tmp = 2.0 / ((k / l) * (((k * t) / l) * (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.2e-114], N[(2.0 / N[(N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e-114

   1. Initial program 34.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 66.3%

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

   9. Applied rewrites 78.1%

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

   if 6.2e-114 < k

   1. Initial program 23.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 63.4%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k \cdot t}{\ell} \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\cos k \cdot \ell\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= (* l l) 5e-306)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ 2.0 (/ k (* (/ l (* (* (* k t) k) k)) (* (cos k) l)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-306) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (k / ((l / (((k * t) * k) * k)) * (cos(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) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if ((l * l) <= 5d-306) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / (k / ((l / (((k * t) * k) * k)) * (cos(k) * l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-306) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (k / ((l / (((k * t) * k) * k)) * (Math.cos(k) * l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if (l * l) <= 5e-306:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / (k / ((l / (((k * t) * k) * k)) * (math.cos(k) * l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-306)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(k / Float64(Float64(l / Float64(Float64(Float64(k * t) * k) * k)) * Float64(cos(k) * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if ((l * l) <= 5e-306)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / (k / ((l / (((k * t) * k) * k)) * (cos(k) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-306], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k / N[(N[(l / N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999998e-306

   1. Initial program 27.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 59.0%

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

   9. Applied rewrites 93.0%

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

   if 4.99999999999999998e-306 < (*.f64 l l)

   1. Initial program 31.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 68.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\cos k \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{k}{\ell}\right)}^{-2}} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(\frac{k}{\ell} \cdot t\right) \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.1e-163)
   (/ 2.0 (* (/ k (pow (/ k l) -2.0)) (* k t)))
   (/ 2.0 (* (* k k) (* (* (/ k l) t) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.1e-163) {
		tmp = 2.0 / ((k / pow((k / l), -2.0)) * (k * t));
	} else {
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (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 <= 5.1d-163) then
        tmp = 2.0d0 / ((k / ((k / l) ** (-2.0d0))) * (k * t))
    else
        tmp = 2.0d0 / ((k * k) * (((k / l) * t) * (k / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.1e-163) {
		tmp = 2.0 / ((k / Math.pow((k / l), -2.0)) * (k * t));
	} else {
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (k / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.1e-163:
		tmp = 2.0 / ((k / math.pow((k / l), -2.0)) * (k * t))
	else:
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (k / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.1e-163)
		tmp = Float64(2.0 / Float64(Float64(k / (Float64(k / l) ^ -2.0)) * Float64(k * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(k / l) * t) * Float64(k / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.1e-163)
		tmp = 2.0 / ((k / ((k / l) ^ -2.0)) * (k * t));
	else
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.1e-163], N[(2.0 / N[(N[(k / N[Power[N[(k / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0999999999999999e-163

   1. Initial program 33.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 76.8%

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

   if 5.0999999999999999e-163 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 53.6%

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

   9. Applied rewrites 60.5%

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

   11. Applied rewrites 62.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\frac{k}{{\left(\frac{k}{\ell}\right)}^{-2}} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(\frac{k}{\ell} \cdot t\right) \cdot \frac{k}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(\frac{k}{\ell} \cdot t\right) \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.1e-163)
   (/ 2.0 (* (* (* (pow (/ k l) 2.0) t) k) k))
   (/ 2.0 (* (* k k) (* (* (/ k l) t) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.1e-163) {
		tmp = 2.0 / (((pow((k / l), 2.0) * t) * k) * k);
	} else {
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (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 <= 5.1d-163) then
        tmp = 2.0d0 / (((((k / l) ** 2.0d0) * t) * k) * k)
    else
        tmp = 2.0d0 / ((k * k) * (((k / l) * t) * (k / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.1e-163) {
		tmp = 2.0 / (((Math.pow((k / l), 2.0) * t) * k) * k);
	} else {
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (k / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.1e-163:
		tmp = 2.0 / (((math.pow((k / l), 2.0) * t) * k) * k)
	else:
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (k / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.1e-163)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / l) ^ 2.0) * t) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(k / l) * t) * Float64(k / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.1e-163)
		tmp = 2.0 / (((((k / l) ^ 2.0) * t) * k) * k);
	else
		tmp = 2.0 / ((k * k) * (((k / l) * t) * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.1e-163], N[(2.0 / N[(N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0999999999999999e-163

   1. Initial program 33.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 77.1%

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

   if 5.0999999999999999e-163 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 53.6%

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

   9. Applied rewrites 60.5%

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

   11. Applied rewrites 62.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(\frac{k}{\ell} \cdot t\right) \cdot \frac{k}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;\ell \cdot \ell \leq 20000000000:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= (* l l) 20000000000.0)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ 2.0 (* (* (/ (* k t) l) (/ k l)) (* k k))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 20000000000.0) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (k * 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) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if ((l * l) <= 20000000000.0d0) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / ((((k * t) / l) * (k / l)) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 20000000000.0) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if (l * l) <= 20000000000.0:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (k * k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (Float64(l * l) <= 20000000000.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) / l) * Float64(k / l)) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if ((l * l) <= 20000000000.0)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 20000000000.0], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e10

   1. Initial program 28.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 85.8%

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

   if 2e10 < (*.f64 l l)

   1. Initial program 32.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 54.9%

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

   9. Applied rewrites 59.0%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 60.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 20000000000:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.6e+67)
   (/ 2.0 (* (* (/ (* k k) (* l l)) (* k k)) t))
   (* (* (/ l (* (* k k) t)) l) -0.3333333333333333)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.6e+67) {
		tmp = 2.0 / ((((k * k) / (l * l)) * (k * k)) * t);
	} else {
		tmp = ((l / ((k * k) * t)) * l) * -0.3333333333333333;
	}
	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 <= 4.6d+67) then
        tmp = 2.0d0 / ((((k * k) / (l * l)) * (k * k)) * t)
    else
        tmp = ((l / ((k * k) * t)) * l) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.6e+67) {
		tmp = 2.0 / ((((k * k) / (l * l)) * (k * k)) * t);
	} else {
		tmp = ((l / ((k * k) * t)) * l) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.6e+67:
		tmp = 2.0 / ((((k * k) / (l * l)) * (k * k)) * t)
	else:
		tmp = ((l / ((k * k) * t)) * l) * -0.3333333333333333
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.6e+67)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / Float64(l * l)) * Float64(k * k)) * t));
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(k * k) * t)) * l) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.6e+67)
		tmp = 2.0 / ((((k * k) / (l * l)) * (k * k)) * t);
	else
		tmp = ((l / ((k * k) * t)) * l) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.6e+67], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5999999999999997e67

   1. Initial program 32.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 65.6%

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

   if 4.5999999999999997e67 < k

   1. Initial program 23.6%

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

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

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

      11. +-rgt-identity N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. associate-*l/ N/A

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

   4. Applied rewrites 41.8%

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

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 52.9%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell\right) \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

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 / ((k * k) * (((k / l) * t) * (k / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 65.2

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

5. Applied rewrites 65.2%

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

7. Applied rewrites 61.7%

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

9. Applied rewrites 68.9%

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

11. Applied rewrites 71.0%

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

12. Final simplification 71.0%

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

Alternative 12: 72.7% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 65.2

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

5. Applied rewrites 65.2%

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

7. Applied rewrites 61.7%

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

9. Applied rewrites 68.9%

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

10. Taylor expanded in t around 0

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

12. Applied rewrites 69.8%

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

13. Final simplification 69.8%

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

Alternative 13: 30.4% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* (/ l (* (* k k) t)) l) -0.3333333333333333))

C

double code(double t, double l, double k) {
	return ((l / ((k * k) * t)) * l) * -0.3333333333333333;
}

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 / ((k * k) * t)) * l) * (-0.3333333333333333d0)
end function

Java

public static double code(double t, double l, double k) {
	return ((l / ((k * k) * t)) * l) * -0.3333333333333333;
}

Python

def code(t, l, k):
	return ((l / ((k * k) * t)) * l) * -0.3333333333333333

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / Float64(Float64(k * k) * t)) * l) * -0.3333333333333333)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / ((k * k) * t)) * l) * -0.3333333333333333;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. associate--l+ N/A

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

   10. metadata-eval N/A

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

   11. +-rgt-identity N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/r* N/A

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

   15. associate-*l/ N/A

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

4. Applied rewrites 43.3%

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

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 35.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

8. Taylor expanded in k around inf

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

10. Applied rewrites 26.5%

    \[\leadsto \left(\ell \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right) \cdot \color{blue}{-0.3333333333333333} \]

11. Final simplification 26.5%

    \[\leadsto \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell\right) \cdot -0.3333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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))))